Mathematical Analysis and Numerical Methods for Science and Technology || Spectral Theory

Chapter VIII. Spectral Theory

Introduction

In volumes 1 and 2 we studied the problem: find u satisfying

(1) .1u = f in Q c IR"

+ boundary conditions (for example u/ao = uo) and more generally

(2) Au = f in Q c IR"

+ boundary conditions

where A is a differential operator depending only on the variable x E IR". We solved such problems in Chap. VII. We also showed in Chap. IB that, besides problem (2), other applications lead to the study of the solution of the following problem: find u satisfying

(3) .1U+AU = f in Q c IR"

+ boundary conditions (for example u/ao = uo);

or more generally

(4) Au + Au = f In Q c IR"

+ boundary conditions.

In these problems, A,f and Uo are given and A is a real or complex number given or not. This type of problem possesses a solution only for certain values of the parameter A, called eigenvalues of the Laplacian .1 (resp. of the operator A) in the equation (3) (resp. (4)). We have seen elsewhere in Chap. I that these values of A have a physically important meaning (for example in Chap. lA, §6, where we had f == 0 and Q = IR", and A was the Hamiltonian of the quantum system, the values Ak are the energy levels of the system). Likewise, we have encountered equation (4) in Chap. lA, §1, in elasticity (vibrations of a membrane) and in Chap. lA, §4 in electromagnetism. The solution of (2) (or of (4)) involves the study of the properties of the linear operator A. Let us suppose that the latter maps a space E into F, and has a domain D(A) c E.

R. Dautray et al., Mathematical Analysis and Numerical Methods for Science and Technology© Springer-Verlag Berlin Heidelberg 2000

2 Chapter VIII. Spectral Theory

If E and F have finite dimension, the theory of linear operators from E into F reduces to the theory of matrices. If E = F, there then exists a finite number of eigenvalues Ak and eigenvectors Uk such that

(5)

which can be simple or multiple. The generalisation of these properties to spaces of infinite dimension, without losing the geometric properties of euclidean spaces, can readily be made when E and F are Hilbert spaces (spaces particularly well adapted to the study of the linear problems of physics and mechanics) and when A is a bounded operator. For example, if E = F = L 2(Q) and the operator A is bounded, compact and selfadjoint, then the study of (5) becomes quite simple (see §2) and allows (4) to be solved by means of the Fredholm alternative. If A is self-adjoint, unbounded and positive in L 2 (Q), with compact inverse, analogous results are again obtained. Finally, we shall see how one can treat the most general case and the results obtained. The spectrum of an operator does not contain all the information that characterises the operator A. This leads to the introduction of the spectral family for A, which, in its case, completely characterises the operator A.

§1. Elements of Spectral Theory in a Banach Space. Dunford Integral and Functional Calculus

Let X be a complex Banach space with norm denoted by I Ix; 5£(X) denotes the space of continuous linear mappings of X into itself or equivalently the space of operators bounded on X.

Equipped with the norm II A II = sup IAxl x , 5£(X) is a Banach space. Now let A Ixlx';; I

be an unbounded operator on X, with domain D(A); we write

(1.1) AA = AI - A, A E C, I the identity on X .

The study of the set 9f A E C for which A). is invertible, and of the properties of A; 1

whenever this operator exists, constitutes what is called the spectral theory of the operator A.

1. Resolvant Set and Resolvant Operator. Spectrum of A

We introduce the following notions.

Definition 1. 10 The set of )0 E C such that:

§ 1. Elements of Spectral Theory 3

{i) A;.(D(A)) (the image of D(A) under A;.) is dense in X;

(1.2) ii) A; 1 exists and is continuous from A;.(D(A)) (equipped with the topology induced by X) into X ;

is called the resolvant set for A, denoted by p(A). 2° We write

(1.3) R(A, A) = Ai 1 = (AI - A)-l ;

R(A, A) is called the resolvant operator or the resolvant of A.

When there is no possibility of confusion, we shall write R(A) in place of R(A, A) to denote the resolvant of A.

Definition 2. We denote by u(A) the complement in C of the set p(A), and call u(A) the spectrum of A. We note that u(A) is the union of three disjoint sets denoted respectively by up(A), uc(A), ur(A) where:

-up(A) = {A. E C; A;. is non-invertible} up(A) is the point spectrum of A

-l1c(A) = {A. E C; Ail is unbounded on X, with domain dense in X} I1c (A) is the continuous spectrum of A

-l1r(A) = {A. E C; Ail exists, with domain not dense in X} ur(A) is the residual spectrum of A

Remark 1. In the case where A is a closed operator with domain D(A), a characterization of p(A) is given by

Proposition 1. If A is closed, then

(1.4) p(A) = {A. E C; R(A) = A; 1 E 2(X)} .

Proof If A E p(A), then the domain of R(A) is D(R(A)) = A;.(D(A)) which is dense in X and, from the continuity of R(A), there exists a constant c > 0 such that we have

(1.5) I(AI - A)xlx ~ clxl x for all x E D(R(A)) .

We must show that

(1.6) D(R(A)) = X .

Since D(R(A)) is dense in X, for each y E X, one can find Xn E D(A) such that

A;.(xn ) -+ Y in X

It follows from (1.5) that the sequence Xn converges to an element x in X. The operator A being closed, the same holds for A;. so that we have:

x E D(A) and A;.(x) = y .

Whence (1.4). o


Remark 2. A necessary and sufficient condition for A E O'p(A) to hold is that the equation

(1.7) Ax = AX·

admits at least one solution x#-O in X. In this case, A is called an eigenvalue of A, and a solution of(1.7), an eigenvector of A. The set of eigenvectors relative to an eigenvalue A is a vector subspace of X which is none other than the kernel E). of A).. It is called the eigenspace associated with A; the dimension of E). is called the multiplicity of the eigenvalue A. 0

Example 1. Finite dimensional case. Suppose that the dimension of X is finite. 1 An operator A E .st(X) is represented with respect to a basis of X by a matrix (aij)' It is known that the eigenvalues are obtained as roots in C of the characteristic polynomial P(A) = det(Abij - aij)' where det(aij) denotes the determinant of the matrix (aij) and bij the Kronecker symbol (thus there can be neither a continuous spectrum nor a residual spectrum). We further note (Cayley-Hamilton theorem) that A satisfies the algebraic equation P(A) = O. 0

Example 2. Example of a continuous spectrum. Let X = L2(1R), the space of (classes of) functions square integrable with respect to Lebesgue measure on IR with values in C. Equipped with the norm

X is a Hilbert space. Let us consider A to be the operator of multiplication by x, which corresponds in quantum mechanics to the "position" operator of a particle2 •

We thus have Af = x. f: x r--. xf(x) where A is an unbounded operator in X defined on

Hence, we have:

(1.8)

D(A) = {fE X; x.fE X}.

{

i)

"') .~~) III

O'p(A) = 0 = O'r(A)

O'c(A) = IR

p(A) = C\IR.

In effect, the condition (AI - A)f = 0 implies

(A - x)f(x) = 0 a.e. and hence f(x) = 0 a.e. ;

it follows that O'p{A) = 0.

1 Recall (see Chap. VI) that in a finite dimensional space, all linear operators are continuous. 2 See Chap. lA, §6.


For)' E IR, R()') = (AI - A) -1 is defined in particular on

{cp E L 2 (IR); cP = ° in a neighbourhood of x = A} .

We note that if cP E U(IR), the sequence CPn defined by

X _ { cp(x) a.e. x ¢ (A - ~'A + ~)

(1.9) cpn( ) - [1 1 ] ° a.e. x E A - ~,A + ~

is in D(R()')) and satisfies:

fUi ICPn - cplx = )._~ Icp(xWdx -> ° as n -> +00.

n

It follows that D(R(A)) is dense in X and A E I1c(A). Now, for A E C\IR,3 for all

1/1 E X, the function cP = _1/1_ is in D(A) and satisfies x - A

(x - A)cp = 1/1 .

Thus D(R(A)) = X for A E C\IR. Furthermore, it is easily verified that A is closed, so that, by Proposition 1, p(A) = C\IR, whence (1.8). 0

Example 3. Example of a residual spectrum. Let X = [2(C), the space of square summable sequences with values in a::::. If u E X, then u = (un)n EN with

L IUn l2 < + 00. n E N

( )1/2

Equipped with the norm lulx = L IUn l2 , X is a Hilbert space. nEN

Let A be defined on X by:

. {Va = ° Au = v, v = (vn)n EN wIth _ 1 Vn - un - 1 ,n ~ .

Then ° E I1r(A) because A(X) is not dense in X being orthogonal to u = (un)n EN

(U o = 1, Un = 0, '<In ~ 1). 0

Example 4. Example of an empty spectrum. Letl = ] 0, 1 [, X = U (I ) and A the derivative operator in the distribution sense:

def du Au = dt'

3 We use the notation C\IR to denote the set (also denoted by C -IR) consisting of the complex plane with the real axis removed.


with domain

def {dU } D(A) = U E L2(I); dt E L2(I), u(O) = 0 .

We observe that for given f E U(I) and A E C, the equation

du dt - Au = f

admits one and only one solution

u(t) = J: e;'(t - S)f(s) ds

in the space D(A). Thus we have p(A) = C and the spectrum of A is empty. 0

Example 5. Resolvant set and resolvant of a self-adjoint operator in a Hilbert space. Let X be a complex Hilbert space; we denote by ( , h the scalar product in X.

Proposition 2. Let A be a self-adjoint operator with domain D(A). Then the following properties hold:

i) p(A) ::J C\IR, whence O'(A) c IR and the spectrum of A is real;

1 ii) II R(A) II ~ IlmAI' A E C\IR (1m: imaginary part of A);

iii) Im(A;.x, xh = (1m A) lxii-for all x E D(A).

Proof If x E D(A), then (Ax, xh E IR because (Ax, xh = (x, Axh = (Ax, xh, and part iii) follows. Thus

IlmAI.lxli ~ I(A;.x, xhl ~ IA;.xlx·lxlx,

from which we deduce

(1.10) IA;.xlx ~ IlmAI·lxlx ,

so that R(A) = A,l" 1 exists for 1m A #- O. Furthermore,

(1.11) A).(D(A)) is dense in X if 1m A#-O .

In effect, suppose that there exists y#-O orthogonal to A;.(D(A)). Then we have (Ax, yh = A(X, y) for all x E D(A), whence y is such that the function x -+ (Ax, xh is continuous on D(A) in the topology on X. Thus y E D(A) and

(x, A;:y) = 0 for all x E D(A) .

Since D(A) is dense in X, we have A;:y = 0, hence Ay = Iy which contradicts the fact that (Ay, yh is real. Thus if y is orthogonal to A;.(D(A)), then y=O and A;.(D(A)) is dense in X. Since, moreover, A is closed, Proposition 1 together with (1.10) establishes parts i) and ii). 0


2. Resolvant Equation and Spectral Radius

We now return to the general case where X is a Banach space.

Theorem 1. Let A be a closed linear operator with domain D(A) in a Banach space X. Then: i) the set p(A) is open in the complex plane C; ii) the function A H R(A.) = R(A, A) is an holomorphic function of A, in each connected component of p(A).

Proof From Proposition 1, for all A E p(A), R(A) E .5l'(X). Let Ao E p(A); consider the series:

(1.12) S(A) = R(AO{ I + "~1 (AO - A)"[R(Ao)]" ]

which converges in the norm of .5l'(X) whenever

(1.13) IA - Aol·11 R(Ao)ll < 1 .

In the open disc in the complex plane defined by (1.13), the series defined by (1.12) determines an holomorphic function of A. By writing U - A = (A - Ao)1 + (Aol - A), we see that for each A in the disc defined by (1.13):

(U - A)S(A) = S(A)(A./ - A) = I ,

so that in this open disc, S(A) coincides with R(A). Thus, we have shown that for each Ao E p(A), p(A) contains an open disc in which the function A -4 R(A) is holomorphic. Theorem 1 then follows. 0

Theorem 2. If A and p. E p(A) and if R(A) and R(p.) are in .5l'(X),4 then R(A) and R(p.) satisfy 'the resolvant equation':

(1.14) R(A) - R(p.) = (p. - A)R(A)R(p.) .

Proof In effect, we can write:

R(A) = R(A)AIlR(p.) = R(A)[(P. - A)I + (U - A)]R(p.)

= (p. - A)R(A)R(p.) + R(p.),

from which (1.14) follows.

For the present we consider the particular case of a bounded operator.

Theorem 3. If A E .5l'(X), then: i)

(1.15) lim II A" 11 1/" = ra(A) "~ 00

exists;

4 Note that if A is closed, R(l) E .P(X) whenever l E p(A) in the light of Proposition 1.

o


ii) r a(A) is called the spectral radius of A and we have:

(1.16) rAA) ~ IIA II ;

iii) iflAI > rAA), then R(A) exists and is given by

00

(1.17) R(A) = L A- n A n - I ,5

n = I

the series converging in 2'(X).

n ~ 1

We must show that

(1.18)

For this we note that, for all 0 > 0, an m E N* can be found such that

IIAmlil/m ~ r + 0.

Then let n E N*; we can write n = mp + q, ° ~ q ~ m - 1, and since II A . B II ~ IIA II II B II, we get:

IIAnlil/n ~ IIAmllplnllAllqln ~ (r + o)pmln.IIAll qln .

Since pm/n ---+ 1 when n ---+ + 00 and q/n ---+ ° under the same conditions, we have:

(1.19) for all 0 > 0, lim IIAnlil/n ~ r + o.

From (1.19) we deduce (1.18) and part i) of the Theorem. Furthermore, since II An II ~ II A II n, we have:

lim IIAnlil/n ~ 1lA11, whence(1.16).

If IAI ~ rAA) + 0,0 > 0, we have for large enough n:

( ro(A) + ~)(n -I)

IIA-nAn-111 ~ , (ra(A) + o)n

which shows that the series (1.17) converges in 2'(X). Finally, by multiplying the second member of (1.17) on the left and on the right by A.I - A, an easy calculation shows that we obtain I so that the sum of the series represents R(A). Theorem 3 then follows. 0

Remark 3. 1 ° From Theorem 3, there follows immediately:

(1.20) for A E 2'(X) , p(A) of- 0 .

2° In this case the value of ra(A) can be made more precise. We have

5 With the convention AD = l.

§ 1. Elements of Spectral Theory

Theorem 4. For A E 2(X), we have:

i)

(1.21 ) ra(A) = sup IAI; A E a(A)

00

ii) the series LA-"A" diverges iflAI < r.,.(A). o

Proof First of all, it follows from Theorem 3 that we have

ra(A) ~ sup IAI, A E a(A)

9

so that it only remains to show that ra(A) ~ sup IAI in order to obtain (1.21). AEa(A)

We note, from Theorem 1, the result that R(A) is an holomorphic function of A,

whenever IAI > sup 1111. Il Ea (A)

Thus R(A) has a unique representation as a Laurent series in positive and negative

powers of A, which converges in 2(X) for IAI > sup 1111. IlEa(A)

From the conclusion iii) in Theorem 3, this Laurent expansion must coincide with

the series in the second member of (1.17) for IAI > sup 1111. IlEa(A)

Since lim II A -" A" II = 0 for IAI > sup 1111, we shall also have for all e > 0 and n --Jo 00 1.t E n(A)

for all large enough n:

IIA"II ~ [e + sup IIlIJ"; Il Eu(A)

so that

(1.22) r.,.(A) = lim IIA"II I !n ~ sup 1111, n ~ 00 IlEa(A)

from which follows part i) of Theorem 4. To establish part ii), let us put:

(1.23) p = inf{r ~ 0; the series (1.17) converges in 2(X) for IAI > r} ;

then for I A I > p, lim II A - " A" II = 0 so that, exactly as in the proof of (1.22), we

have

(1.24) lim ·IIA"II I !n = ra(A) ~ p, "~ 00

whence Theorem 4. o


3. Dunford Integral and Operational Calculus

Referring still to a Banach space X and a bounded operator A E .!.e(X), we denote by J'l'(A) the set of all complex valued functions holomorphic in a neighbourhood (not necessarily connected) of the spectrum O"(A) of A. The neighbourhood in which f E J'l'(A) is defined can depend onf Letf E J'l'(A) and let () be an open superset of O"(A) contained in the domain of holomorphy off We assume that the boundary o() = r of () consists of a finite number of rectifiable Jordan curves, each positively orientated, and we denote by r h the contour thus orientated. The formula, analogue of the Cauchy formula for scalar holomorphic functions,

(1.25) f(A) = -21 . r f(l)R(l)dl m Jr~

defines an operatorf(A) E .!.e(X) and the integral on the right hand side of (1.25) is given the name Dunford integral. From the theory of the Cauchy integral, the operator f(A) depends only on the functionfand the operator A, but not on the open set () involved. N. Dunford has developed a functional calculus which goes under the name of the Duriford operational calculus,6 the elementary rules of which are given in

Theorem 5. Let A be a bounded operator (A E .!.e(X» in a Banach space X. 10 Let f, g E J'l'(A), (x, fJ E C, then i) (Xf + fJg E J'l'(A) and (Xf(A) + fJg(A) = «(Xf + fJg)(A) ii) f. g E J'l'(A) and f(A).g(A) = (f. g)(A).7 20 Iff E J'l'(A) has a Taylor series representation

(1.26)

in a neighbourhood V of O"(A), then we have

00

(1.27) f(A) = L anAn in the sense of .!.e(X) . o

30 Let (j,;)n E N be a sequence of functions f" E J'l' (A); suppose further that,for all n, the f" are holomorphic in a fixed neighbourhood V of O"(A). Then

{iff" ~ f uniformly on V ,

(1.28) thenf,,(A) ~ f(A) in .!.e(X) .

Proof 10 Part i) is immediate. We prove part ii). Let (!Ji (i= 1, 2) be two open neighbourhoods of O"(A) such that r i = O(!Ji (i = 1,2) consists of a finite number of rectifiable Jordan curves. Suppose that (!J 1 uri c (!J 2 and that (!J 2 U r 2 is in the domain of holomorphy off and g.

6 See Dunford-Schwartz [1] volume I. 7 We denote by either the multiplication sign (.) or no sign at all, the composition (0) of two operators in 2'(X).

§ 1. Elements of Spectral Theory

We have

f(A)g(A) = - ~ i f(A)R(A) dA i g(J.L)R(J.L) dJ.L 4n r, r2

making use of the resolvant equation (1.14), we then deduce:

f(A)g(A) = - ~ i f(A)g(J.L) [R(A) - R(J.L)] dA dJ.L , 4n r, x r2 J.L - A

whence

f(A)g(A) = -21 . i f(A)R(A) {-21 . i g~) A dJ.L} dA m r, m r2 J.L

1 i { 1 i f(A) } - -2' g(J.L)R(J.L) -2' ----=--1 dA dJ.L. m r2 m r, J.L

But from the properties of the ordinary Cauchy integral:

~ i g(J.L) dJ.L = g(A), (A E rd , 2m r2 J.L - A

so that

f(A)g(A) = ~i f(A)g(A)R(A)dA = (f.g)(A); m r,

whence part ii).

Fig. I Fig. 2

11

2° Let r,,(A) be the spectral radius of A; by hypothesis, V contains a closed disc DE = {A. E C; IAI ~ r.,.(A) + e}, e > 0 sufficiently small, on which the series 00

L anAn converges uniformly. o


Furthermore, R(A) has the Laurent expansion

00

R(A) = LA - nAn - 1 (Theorem 3) . 1

Hence, if r, = aD, with positive orientation:

f(A) = -. L an L An- k A k - 1 dA = L anAn. 1 00 00 i 00

2nl n = 0 k = 1 r, n = 0

3° (1.28) follows immediately from (1.25). o

Theorem 6 (Dunford spectral theorem or 'Spectral Mapping Theorem'). Let A be a bounded operator (A E 2'(X)) in a Banach space X. Iff E £(A), then

(1.29) f(a(A)) = a(f(A)) .

Proof. Let A E a(A) and let g be the function defined by

g(Jl) = f(A) - f(Jl) for Jl -:f A A - Jl

and g(A) = f'(A) .

From Theorem 5:

f(A)I - f(A) = (U - A).g(A);

it follows that if f(A)I - f(A) is invertible with inverse BE 2'(X), then G(A).B E 2'(X) is the inverse of U - A. Thus A E a(A) implies f(A) E a(f(A)), hence f(a(A)) c a(f(A)). Suppose that A E a(f(A)) with A ¢ f(a(A)). Then the function qJ(Jl) = (f(Jl) - A) -1 is in £(A) and we have qJ(A)(f(A) - U) = I which contradicts the hypothesis that A E a(f(A)), whence Theorem 6. 0

Theorem 7. Let A be a bounded operator (A E 2'(X)) in a Banach space X. Letf E £(A) and g E £(f(A)). Then, ifh = g of, h E £(A)) and h(A) = g of(A).

Proof. The fact that h E £(A) follows from Theorem 6. Denote by 0 1 an open neighbourhood of a(f(A)) of which the boundary r 1 is the union of a finite number of rectifiable Jordan curves, and such that 0 1 u r1 is included in the domain of holomorphy of g. Let O2 be an open neighbourhood of a(A) of which the boundary r 2 is also the union of a finite number of rectifiable Jordan curves, such that O2 u r2 is included in the domain of holomorphy off and suppose that

f(02 u r 2) c 0 1 .


Then, for A E r 1 we have

1 r 1 R(A,J(A)) = 2ni Jr, A _ f(Jl) R(Jl, A)dJl,

g(f(A)) = -21 . r g(A)R(A,J(A))dA nl Jr,

1 1 g(A) = - -2 A _ f( ) R(Jl, A)dJldA 4n r, x r, Jl

= f r R(Jl, A)g(f(Jl)] dJl = h(A) , 1tI Jr,

and from the Cauchy integral, we have

1 r g(A) g(f(Jl)) = 2ni Jr, A - f(Jl) dA ,

whence Theorem 7. o

4. Isolated Singularities of the Resolvant

Let A be a closed linear operator in a Banach space X and Ao an isolated singularity of the resolvant R(A); then R(A) can be expanded in a Laurent series, in a neighbourhood of Ao, in the form:

+00

(1.30) R(A) = L (A - Ao)" An , n = - o:x)

with

(1.30)' 1 r R(A)

An = 2ni Jc;o (A _ Ao)" + 1 dA ,

where C l' is the boundary, with direct orientation, of a disc, of radius sufficiently small that only the singularity Ao lies in its interior. We then have

Proposition 3. The An defined by (1.30)' are bounded linear operators which commute amongst themselves and are such that

i) AAkx = AkAx for all x E D(A), k E lL., ii) AkAm = 0 for all k ~ 0, m ~ -1,

iii) An = (-1)" A n+ 1 for all n ~ 1, iv) A_ P _ q + 1 = A_p.A_ q for all p, q ~ 1, v) A _ q = (A _ 2 )q - 1 for all q ~ 2.

Proof The mutual commutativity property of the An and the fact that the An are


bounded linear operators on X follow from the Cauchy integral properties (1.30)'. Part i) follows similarly. To verify ii), we substitute the expression for R(A) given by (1.30) into the resolvant equation R(A) - R(J1.) = (J1. - A)R(A)R(J1.); we thus obtain

~oo Ak (A - Ao)k - (J1. - Aot __ _ ~oo (1.31) L. L. AkAm(A - Ao)k(J1. - Ao)m,

- 00 (A - Ao) - (J1. - Ao) k. In = - 00

and can assert that the terms of the form (A - Ao)k(A. - Ao)'" are absent from the series expansion on the left hand side for k ~ 0 and m ~ - 1, whence ii). Furthermore, the identification of terms in (A - A.O)k(A - Ao)'" in (1.31) gives iii) and iv) (for the details, see Yosida [1], p. 228); v) follows easily from iv). 0

We note also

Proposition 4. We have i) An = (A - A.oJ)An+ 1, n ~ 0 ii) (A - AoJ)A-n = A-(n+I) = (A - AoJ)nA_I,n ~ 1

iii) (A - Aol)Ao = A_ 1 - I.

Proof The integral representation of An shows that An(X) c D(A) so that Proposition 4 follows from the identity:

+00 +00

I = (A.J - A) L An(A - Ao)" = {(A - Ao)1 + (Aol - A)} L Ak(A - Ao)k -00 k=-oo

= L (A. - Ao)k + 1 Ak + L (Aol - A )Ak(A - Ao)k . keZ keZ

def def def . Thus putting P = A-I' D = A-2' S = -Ao, the expanSIOn (1.30) can be

written:

00

(1.31)' L (A. - A.o)"sn+I; n=O

the operator P is the projection on Xo = PX along Xl = (I - P)X; thus X = X 0 EEl Xl' and the operator A, commuting with P, leaves stable the subspaces Xo and Xl, i.e. putting D(A(i») = Xi n D(A), i = 0,1, we have

(') def A I X = Ax E Xi' Vx E D(A(i») , i = 0,1, D(A) = D(A(O») EEl D(A(I») ,

A = A(O) EEl A(1) and R(A, A) = (A.J - A)-l

= R(A, A (0») EEl R(A, A (1»), A =f:. Ao .

Denoting again by R(A, A (0») (resp. R(A, A (1»)) the linear extension of R(A, A (0») (resp. R(A, A(1»)) to X by 0 on Xl (resp. X o), we write:

R(A) = R(A, A) = R(A, A(O») + R(A, A(I») ,

§ I. Elements of Spectral Theory

and we can verify, with the aid of Propositions 3 and 4, that:

with

(1.31)" D = (A - AoI)P = (A(O) - AoI)P, (A - AoI)S = I - P,

thus

(and hence SP = PS = O,DS = SD = O,D = DP = PD).

We now demonstrate

15

o

Theorem 8. If Ao is a pole of order m of R(A), then Ao is an eigenvalue of A, and we have

(1.32) {i) A_ l(X) = N[(AoI - An

ii) (I - A_ d(X) = (AoI - A)"(X) for n ~ m,

and hence, in particular:

(1.33) {X = N«AoI - A)") ® (AoI - A)"(X) for n ~ m.

Proof We show (which will prove Theorem 8) that:

{i) N" ~ N«AoI - A)") = Xo

def ii) M" = (AoI - A)"(X) = Xl for n ~ m.

(1.34)

i) By hypothesis, we have Dn = 0, '<In ~ m. Now D" = (A - AO I)" P, hence PX = Xo c N«A - AoI)") = N", '<In ~ m. Furthermore, for all x EN", using (1.31)" we have:

S"(A - AoI)"X = (A - AoI)"S"X = (I - P)x = ° thus tV" c X 0, and i) follows. ii) From i) we have:

M" = (AoI - A)"(X) = (AoI - A)"(Xd = (AoI - A(l))nx l ·

Now (see (1.31")) (AoI - A(l») has an inverse -S = - R(Ao, A(l») on Xl' hence <AoI - A(l»)Xl = Xl' and M" = Xl. The number dim X 0 is called the algebraic multiplicity of the eigenvalue Ao of A. 0


§2. Spectral Decomposition of Self-Adjoint and Compact Normal Operators in a Separable Hilbert Space and Applications

1. Hilbert Sums

1.1. External Hilbert Sum of Hilbert Spaces

Let (Hn)n E 1\1 be a sequence of complex Hilbert spaces; we denote by ( , ). (resp. I In) the scalar product (resp. the norm) in Hn. Let

(2.1 )

It is immediately verifiable that Jf is a complex vector space on which the sesquilinear form

{u, v} 1-+ (u, v).JI'" given on Jf x Jf by

(2.2)

(which exists by virtue of the Cauchy~Schwarz inequality) defines a scalar product. We have

Theorem 1. Equipped with the scalar product ( , )£" Jf is a Hilbert space.

Proof Let (u(P»)p E 1\1 be a Cauchy sequence in Jf. It satisfies:

(2.3) {'tie> 0 there exists P'ooE N; for all p > P. and for all q ~ 0

lu(p+q) - u(P)Ii- = I. lu~p+q) - u~)I; < e2 ,

n~O

from which there follows:

(2.4) for all n E N, the sequence (u~P»)p E 1\1 is a Cauchy sequence in Hn ;

Hn being complete, there exists Un E Hn such that lim lu~P) - unln = O. Putting p- + co

U = (un)n E 1\1, we must show that U E Jf and that u(p) -+ U in Jf as p -+ + 00.

From (2.3), for all N E N, we have:

§2. Spectral Decomposition of Compact Operators

Letting q ~ + 00 in the finite sum in the first member in (2.5), we obtain:

N

L IUn - u~P)I; < [;2, p > p.. for all N EN, o


00

(2.6) L IUn - u~P)I; < [;2, p > p, . o

Let v(p) = (v~P»)n E N' v~P) = un - u~P). From (2.6), we have

(2.7) {i) v(p) = U - u(p) E ye

ii) v(p) ~ 0 in ye as p ~ + 00 ,

whence Theorem 1.

Now let rk be the mapping of Hk ~ ye, (k EN), defined by

(2.8) { J1. E Hk , rk (J1.) = (un)n EN

(b~ Kronecker symbol) ;

it is immediately verifiable that

with Un = J1.b! for all n ,

{i) rk E 2(Hk' ye)

(2.9) ii) r k is an isometry of H k onto a closed subspace of ye .

17

o

It is thus possible to identify Hk and rk(Hk), so that Hk can be regarded as a closed subspace of ye. In what follows, we assume that this identification has been made; we then have

Proposition 1. i) For p =I q, H P and H q are orthogonal.

ii) For all U = (un)n ENE ye, we have U = L Un' n EN

iii) If Pn E 2(ye, Hn) is the orthogonal projection of ye on Hn we have for all U = (un)n ENE ye, Un = PnU.

iv)

(2.10) { U Hk} = ye kEN

where { U Hk} is the vector space generated by the union of the Hk· kEN

Proof i) If J1. E Hp, v E Hq , p =I q, we have

(J1., v)Jf' = (rp(J1.), rq(v))Jf' = 0 .

ii) We need to show that if U = (un)n ENE ye then p p

vp = L rn(Un) = L Un ~ U in ye as p ~ + 00 . n=l n=l


For this, we need to show that Ivp - ulJF -+ 0 as p -+ + 00, or what comes to the same thing, to show that

{ i) ~~~ w)JF -+ (u, w)JF for all WE Yf ,8

ii) IVplJF -+ lulJF as p -+ + 00 .

(2.11)

If W = (Wn)n ENE Yf we have:

(vp, w)JF = C to rn(un), W ) JF = n to (un' wn)n

from the definition of rn; so that, since

we have established part i) of (2.11). Moreover,

IVpl~ = Intorn(Un)l: = nto lunl; -+ n~Nlunl;, whence part ii) of (2.11). iii) In identifying rn(un) and Un' we have

OCJ

U = L Un' n=O

so that Pn E 5f(Yf, Hn) for all n; then

OCJ

Pn(u) = L Pn(up) = Pn(un ) = Un . p=O

iv) From ii), it is evident that the vector subspace generated by the union of the H n

is dense in Yf; but Yf is also the closed vector subspace generated by the union of the H k • 0

Definition 1. The space Yf is called the external Hilbert sum of the sequence (Hn)n EN and is denoted by

(2.12)

1.2. Direct Hilbert Sum of Orthogonal Subspaces of a Hilbert Space

Definition 2. Let H be a complex Hilbert space. H is said to be the direct Hilbert sum of a sequence (Hn)neN of closed vector subspaces of H if

8 Recall that it is a question of weak convergence in Jff which we denote by vp ~ u as p -+ + 00.

§2. Spectral Decomposition of Compact Operators 19

Ii) for p -:F q, H p and H q are orthogonal, (2.13) ii) the closed vector subspace generated by the union of

the Hn is H .

It is natural to compare Hand Yf = EB Hn whenever Hn is given the topology

induced by that of H. We have

Theorem 2. Let H be a Hilbert space, direct Hilbert sum of a sequence (Hn)n E N of closed subspaces. There exists a unique unitary operator IJIf from H onto the external direct sum Yf such that for all n EN, the restriction IJIfl n oflJlf to Hn coincides with rn defined in (2.8).

Proof a) Uniqueness oflJlf Let Ii be the vector space (c H) generated by the union of the Hn

(Ii = L~NHn}). Let IJIf 1 and IJIf 2 be two unitary operators having the property in question. Then IJIf 1

and IJIf 2, which coincide on each H n , coincide on Ii and consequently on H, since Ii is dense in H. b) Existence of IJIf We have to show that there exists IJIf E !l'(H, Jf) such that

(lJIf(u), lJIf(v)).)f' = (u, v).)f' for all u, v E Yf .

Denote by.if the subspace of Yf formed by the sequences u = (un)nEIIi' UnEHn, all but a finite number of whose elements are null. We then denote by "Y" the mapping of it onto Ii defined by

U E it, "Y"(u) = L Un; n EN

"Y" is a linear mapping, bijective from it onto Ii such that for all u, v E it:

("Y"(u), "Y"(V))H = (L Un' L Vn) = L (Un' Vn)n = (u, V).)f' . n n H n

Since it (resp. Ii) is dense in Jf (resp. H), "Y" extends to a unitary operator, again denoted by "Y", from Yf onto H. It is plain that IJIf = "Y" - 1 has the property in question. 0

Consequence of Theorem 2. Theorem 2 permits the identification of Hand Yf. Henceforth we shall make this identification and will write:

(2.14) H = EB Hn· n EN

From Proposition 1 there then follows


Proposition 2. Let H = EEl H n be the direct Hilbert sum of the closed subs paces • EN

Hn of H. For n E N, let Pn be the orthogonal projection of H onto Hn. Then: 1 ° for all u E H

(2.15) u = L Pnu, lul~ = L I Pnul~ ; "EN nEN

20 if {Un}. E N is a sequence from H, such that Un E H n "In and I I Un I~ < + 00, • EN

then the series L un converges in IHI to an element u such that Un = Pn u for "E N

all n E N; 30 for all u, v E H, we have:

(2.16) (u, V)H = I(Pnu, PnV)H' n

Remark 1. Finally, recall that if dim Hn = 1, then Hn = {lCen }. If H = EB Hn ,

"E N then {en} nE iii constitutes an orthonormal basis of H if II en II = 1. Similarly, if the H n

are fini te dimensional, by choosing an orthonormal base in each H n' the set of these bases constitutes an orthonormal basis of H. 0

2. Spectral Decomposition of a Compact Self-Adjoint Operator

For the moment we consider the case where X is a normed vector space. We recall that an operator A E 2(X) is compact if the image A(B) of the unit ball of X under A is relatively compact. 9

We also recall the following result due to Riesz and for which we refer to Yosida [1], p. 85.

Proposition 2' (Riesz). Every locally compact 10 normed vector space has finite dimension. We then have

Proposition 3. Let A E 2(X) be a compact operator. Let EA denote the kernel of AA = AI - A which is not reduced to {O} whenever A E 0' p(A) (that is to say, the eigenspace associated with A). Then if A E 0' p(A), A =F 0, E A has finite dimension.

Proof Let B be a ball with center O. Its image under A is relatively compact in X. Let A E O'p(A), A =F 0, A(EA) C EA so that if BA = B n EA, BA is a neighbourhood of zero in EA and A(B).) = A.B). is a neighbourhood of zero in EA for A =F 0; since

~- -~

EA is closed in X and A(BA) c A(B) n E). where A(B) is compact, A(B;J is

9 See Chap. VI. §2. 10 In such a space, the bounded sets are relatively compact.


contained in a compact set in E;, for the topology induced by that of X, so that A(B;,) is a relatively compact neighbourhood of zero in E;,. Thus E;, is a locally compact normed space, and hence by Proposition 2' has finite dimension. 0

We now return specifically to the Hilbert space context. We denote by H a complex Hilbert space, and by ( , ) (resp. I I) the scalar product (resp. the norm) in H.

Proposition 4. Let A be an hermitian ll operator on H, that is to say

A E !l'(H) and A = A* .

Then 1 ° A E <T p(A) is real; 2° if A and fJ. E <Tp(A) are such that A '# fJ., then the two eigenspaces E;, and Ep. are

orthogonal.

Proof 1 ° We saw earlier (Proposition 2 in § 1) that p(A) :::> C --'- IR, hence <T p(A) c IR. This can also be seen directly: in effect, if A E <T p(A), there exists U;, E E A, with U;, i= 0; then

Alu;,12 = (Au;" u;,) = (u;" Au;,) = XluAI2

whence A = X. 2° Let UA E EA, up. E Ep., A i= fJ.. Then

A(UA, Up.) = (Au;" Up.) = (UA, Aup.) = fJ.(u;" Up.) ,

whence (uA, Up.) = 0 if A '# fJ.. o Proposition 5. Let A be a compact hermitian operator. Let V(O) be a neighbourhood of 0 in IR, VC(O) its complement. Then VC(O) n <Tp(A) contains only afinite number of points. In other words, <T p(A) is either finite or denumerable and the set of eigenvalues can be arranged in a sequence of real numbers converging to zero:

o E O'p(A) if 0 is an 'eigenvalue', i.e. if the kernel of A i= {OJ .

Proof From Theorem 3 in §1, we know that O'p(A) c [ - IIA II, IIA,,]; suppose that for 0 < e < "A II we can find a sequence {An}n E flJ, An '# An' if n i= n', of eigenvalues of A with

e ~ IAnl ~ II A" .

Since [e, II A II ] is compact in IR, we can extract a subsequence An -+ A '# O. To each An is associated an E;'n and we can take Un E E;'n with luni = 1; from Proposition 4 we note that:

(un,um ) = 0 if n i= m, IUn - um l2 = 2.

der Since Uii E Bl (0) (closed ball with centre 0 and radius 1) Auii' = Yii' E A(Bl (0)), the

II Also called self-adjoint; see Chap. VI, §3.1.2, p. 353.


latter being relatively compact by hypothesis; hence we can extract from {Yii} a sub-

. 1 1 h' sequence Yv -+ Y In H. Since Auv = AvU" then Uv = - Yv -+ - y. But t is IS Av A

impossible since IU n - urnl = J2 and the sequence Uv cannot be a Cauchy ~~ 0

Proposition 6. Let A be a compact hermitian operator. Then (if H i= {O}), there exists at least one eigenvalue A E IT p(A) such that A = II A II.

Proof Recall that

(2.17) II A II = sup I(Au, u)1 . I u I '" 1

To demonstrate Proposition 6, we can suppose that II A II = 1. Thus,

either sup (Au, u) = + 1, or inf (Au, u) = -1. Suppose that we have I u I '" 1 I u I '" 1

sup (Au, u) = + 1 and denote by B 1 (resp. L d the closed ball (resp. sphere) of I u I '" 1 centre 0 and radius 1. Then a sequence Un ELI can be found such that

(Aun , un) -+ 1 as n -+ + 00 .

Since Un E B1 , AUn is in A(Bd which is relatively compact in H by hypothesis, so that it is possible to extract from the sequence Vn = AUn a subsequence Vv -+ v in H. Since

(2.18)

(Aun , un) can tend to 1 only if IAunl -+ 1 so that Ivl = 1. We show that Uv -+ v. Now

(2.19) Iv - uvl2 = Ivl2 + IUvl2 - 2Re(v, u.) = 2 - 2 Re(v, uv ) •

Since I(v,uv) - (v"u')l ~ Iv - vvl.luvl-+ 0 when v -+ +00,

we have

(2.20) lim Re(v, uv ) = lim Re(v" uv ) = 1 .

It follows from (2.19), (2.20) that Uv -+ v. Thus Uv and Vv have the same limit v and since A E 2(H),

(2.21) Av = v,

whence the result. o Proposition 7. Let A be a compact hermitian operator. Then the subspace Eo = N(A) = kernel of A (possibly reduced to {O}) and the eigenspaces {E;.} ;'E<rp(A)

generate a subspace whose closure is H. In other words,


(2.22) H = EB E;,. (direct Hilbert sum of the E;,.) ;"Eap(A)

(0 E O'p(A) if N(A) = Eo =P {O}).

Proof We remark in the first place that

VA E O'p(A) E;,. is stable under A (i.e. A(E;,.) c E;,.) .

Let E be the closure of the vector subspace generated by the E;,.. Then we also have A(E) c E. Denote by F the orthogonal complement of E, then F is also stable under A and the restriction A of A to F satisfies A = A\ A compact. From Proposition 6, A has an eigenvalue and an eigenvector providing that F =I {O}. Now such an eigenvector E F will also be an eigenvector of A. But such an eigenvector is impossible because F is orthogonal to all the eigenvectors of A. Hence F = {O}. 0

We now proceed to summarise the preceding collection of properties and to demonstrate

Theorem 3. Let H be a complex Hilbert space and A E 5l'(H) a compact hermitian operator. Then 10 the spectrum of A is a pure point spectrum and real, i.e.:

(2.23)

and the set of eigenvalues of A is either finite or can be arranged as a real sequence converging to O. Moreover,

o E O'p(A) if Eo = N(A) =I {O} ;

20 for each A = O'p(A) except perhaps for A = 0

(2.24) dim E;,. < + 00;

the subspaces E;,. are pair-wise orthogonal and H is the direct Hilbert sum of the E;,.;

(2.25)

is called the spectral decomposition of H relative to A. 30 Let P;, be the orthogonal projection onto E;" then from (2.25), we have:

(2.26) 1= L P;, ;, E ap(A)

and {P;,} is said to be a resolution of the identity.

Proof The only point remaining to be demonstrated is (2.23) which is a property of compact operators on a Banach space established in

12 For an infinite dimensional Hilbert space. If H is finite dimensional. then a(A) = apiA).


Lemma 1. Let X be a complex Banach space and A E 2(X) a compact operator. Then

(2.27) if A ¢ lTp(A) and A of- 0, A E p(A) .

In other words, there is neither a continuous spectrum, nor a residual spectrum, except perhaps at the origin.

Proof if We first of all show that:

(2.28) for all A E C, A of- 0, A;.(X) = (AI - A)(X) is closed in X .

Since ). of- 0, by changing A into ~ A if need be, we can suppose that A = 1. Thus,

let {xn} be a sequence in X such that Yn = (I - A)xn converges to y. If

sup IXnlx < + 00, then in virtue of the compactness of the operator A, there n E f\J

exists a subsequence Xv such that Axv converges in X. Hence Xv = Yv + Axv converges to x E X and we have Y = (I - A)x. If

sup IXnlx = + 00, put Al = I - A, N(Ad = kernel of Al = {xEX;AI(X) = O} n E f\J and let (Xn = dist(xn, N(A I »; we can find Wn E N(Ad such that

° ~ (Xn ~ IXn - wnl x ~ (1 + ~ )(Xn .

Then A I (Xn - W n ) = A I xn and, if the sequence {(Xn} is bounded, it is easy to show (see above) that Y E Al (X).

Consider the case where lim (Xn = + 00 . . o-Cf)

If (xn - wn) h' h . fi I d ° we put Zn = I w IC satIs es znlx = 1 an AIZn -+ as n -+ + 00, Xn - wnl

then a subsequence Zv can be found such that Zv -+ wo, AIZv -+ 0. Thus woEN(Ad

(. fli - A I Xv 10 e ect one can extract Zv ~ wo, A I Zv = -+ ° since A I Xv converges, so

Ixv - wvl that Zv = Azv + Al Zv converges strongly). But if we suppose that uv = Zv - wo, we have:

(2.29)

in the first member of (2.29), the second and third terms are in N(Al) so that we must have:

Since

(V + 1) Ixv - wvl x ~ (Xv -v- ,

there is a contradiction. Whence (2.28). ii) Suppose that Ao ¢ lTp(A), Ao of- 0.

uv -+ ° ,

Then A;.o is a bijection from X onto A;.o(X) which, from i), is closed. It then follows


from a theorem of Banach l3 that Aiol = R(Ao) is in 2'(A;.o(X), X). It remains to show that Ao E p(A), that is to say that A;.o(X) = X. Suppose that A;.o(X) # X. Put Xl = A;.o(X), Xn = A;.o(Xn- l ), n ~ 2. Then, from i): Xn+ 1 is a closed subspace of X n, n ~ 0 (Xo = X).

25

Then we can find Un E X n, such that lunl x = 1 and dist(un, Xn + d ~ 1/2. In virtue of the compactness of A, a subsequence {u v } can be extracted from {un} such that Au, converges in X. Now this is a contradiction, since AUn cannot be a Cauchy sequence. In effect, for n > m, we can write:

(2.30)

so that

1 un. m = un + ;:- (A;'oum - A;.oun) E X m + 1 if n > m,

o

which proves Lemma 1 and Theorem 3.

Remark 2. We further note the

o

Proposition 8. 10 Let A E 2'(H), A A *, A compact; (recall that

H = EB E;.) if u = (U;');.eC1 p (A) E H, then: ;'eC1p (A)

(2.31 )

we say that the spectral decomposition of H diagonalises A. 20 Conversely, let A be a finite set in IR, or a sequence of points in IR which converges to 0, such that

(2.32) H = EB H;., dim H;. < + <Xl for A # 0 . ;'eA

Then the operator A defined by

(2.33) Au

satisfies

(2.34) A E 2'(H), A A *, A compact.

13 Banach's "Open Mapping Theorem", see Chap. VI and Dunford-Schwartz [1] or Yosida [I]: Let X and Ybe Banach spaces and T E .!l'(X, Y) a linear operator which is continuous and surjective; then the image under T of every open set in X is open in Y.


Proof 1° Follows immediately from the definition of the E). and from Proposition 2. 2° Conversely, let A be the mapping defined by (2.33). It is trivially linear.

Hun --> u in H then, from Proposition 2, L lu~ - u).1 2 --> 0 as n --> +00. From ).EA

the fact that A is bounded, it follows that

IAun - Aul2 = L A21u~ - u).1 2 ~ M L lu~ - u).1 2 --> 0 ).EA ).eA

as n --> + 00 which shows that A E .5f(H). Furthermore, still making use of Proposition 2 and A c ~, we have

(Au, v) = L (P).Au, P).v) = L (AU)., v).) = L (u)., AV).) ).EA ).EA ).EA

= (u, Av) for all u, v E H . Thus A = A*. We now show that A is compact. For all e > 0 we can write:

(2.35) { A = Al U A2

Al PEA; IAI ~ B}, A2

and hence we can put

PEA; IAI > B}

(2.36) { A = Al + A2

Al U = (AU).)).EA" A2u = (AU).)).EA2

We note that A2 is a finite rank operator because on the one hand A2 is finite and on the other each H). corresponding to A E A2 has finite dimension. Moreover, IIAIII ~ B.

Thus A is the limit in .5f(H) of finite rank operators, and hence is compact (Chap. VI). Naturally, in denoting by E).(A) the eigenspace of A associated with the eigenvalue A, we have E).(A) = H). for all A E O"p(A) = A. 0

Remark 3. 1 ° The theory which we have developed in this §2.2 is also valid for a real Hilbert space H and compact 'symmetric' A E .5f(H) (i.e. t A = A). 2° We note that the proof of Theorem 3 and more particularly the proof of Proposition 6 makes no appeal to d'Alembert's theorem on the solubility of algebraic equations in C (recall that the spectrum of an hermitian operator is real).

3. Spectral Decomposition of a Compact Normal Operator

Definition 3. Let A E .5f (H); A is said to be norma[14 if it commutes with its adjoint A* :AA* = A*A.

14 See Chap. VI.


An hermitian operator A E 2(H) and a unitary operator OU E 2(H) are normal. The normal operators admit a spectral decomposition analogous to that of the selfadjoint compact operators, but here it is essential to consider a complex Hilbert space H. Before stating the generalisation of Theorem 3 to normal operators, we demonstrate

Proposition 9. Let H be a complex Hilbert space and A E 2(H) a normal operator. Then, denoting by E,,(A) the eigenspace associated with A/or each A E ITp(A),15 we have

1° A E IT(A) ¢> ~ E IT(A*), and E,,(A) = E~(A*). 2° For A #- fl, A, fl E ITp(A), E,,(A) is orthogonal to EI'(A). 3° Further, if A is compact, there exists at least one A E IT p(A) with I A I = II A II.

Proof lOA" = AI - A is a normal operator and (A,,)* = ~I - A *. E,,(A) = N(AA) = kernel of AA is stable for (A,,)* since A,( and (A,()* commute and it is easy to verify that the restrictions of A" and (A,,)* to E,((A) are again adjoints. In E,,(A), A" = 0 hence (A~)* = O. Thus Ei(A*) ::) EA(A) and since (Ai)** = A,( it follows that we also have E;.(A) ::) EiA *), whence 1°. 2° If A #- fl, A, fl E ITp(A), u E E,,(A), v E EI'(A) = Eji(A*), we have

A(U, v) = (Au, v) = (u, A*v) = fl(U, v) .

Since A #- fl, (u, v) = O. 3° Now suppose that A is compact normal and II A II = 1. Then A * A is positive hermitian and we have:

II A * A II = II A 112 = 1 .16

On the other hand, A being compact, A* A is the same. From Proposition 6, the eigenspace EdA* A) relative to the eigenvalue 1 is different from {O}. Let F = E 1 (A * A), then F is finite dimensional, stable for A and A * which commute with A * A - I, and the restrictions of A and A * are adjoints of each other and normal. From the known result in a finite dimensional vector space, A acting in F has at least one eigenvalue A (d'Alembert's theorem). Let u). be a corresponding eigenvector; U;. is again an eigenvector of A* for the eigenvalue 1 Then A*Au). = A~U). since A*A - I = 0 on F, hence IAI2 = 1. Thus, as asserted, A has an eigenvalue A, with I A I = 1. 0

The spectral decomposition theorem in this case is then the following:

Theorem 4. Let H be a complex Hilbert space and A E 2(H) a compact normal operator. Then:

def 15 That is to say the kernel of A, = AI - A (see Proposition 3). 16 Inelfect, II A*A II = sup I(A*Au,u)1 = sup IAul' = 1lA11'.

lui ~ 1


1 ° the spectrum u(A) of A is a pure point spectrum, that is to say:

(2.37)

and the set of eigenvalues of A is either finite or can be arranged as a complex sequence converging to 0 in C;

o E up(A) if Eo(A) = N(A) = N(A*) =I {O} ;

2° for each A E up(A) except perhaps for A = 0

(2.38) dim E;.(A)( = dim E~(A *)) < + 00.

If A =I j1, E;.(A) is orthogonal to EI'(A) and H is the direct Hilbert sum of the E;.(A):

(2.39) H = EB E;.(A) ;'EO"p(A)

is the spectral decomposition of H relative to A.

Proof The proof is the same as that for Theorem 3, taking into account Proposition 9. 0

We now establish a result which will be useful to us in what follows.

Theorem 5. Let A be a compact operator from H 1 to H 2 (Hi Hilbert spaces, i = 1, 2). Then A can be decomposed into the form

(2.40) A = UT,

where i) T is a positive compact hermitian operator from H 1 to H 2;

ii) V is a unitary operator from T(H l) to H 2'

Proof Let B = A * A; as has been shown (see Proposition 9), B is pOSItIve hermitian and compact. (A:H1 -+ H2 , A*:H2 -+ H l , so that A*A:H1 -+ H1 is compact positive hermitian.) From the spectral decomposition theorem for compact positive hermitian operators, we can find an orthonormal basis for H 1 formed of eigenvectors {en}nE N* relative to a sequence of eigenvalues {An}nE N0 of B; An ~ O. Let T be defined on each en by:

T(en) = fin en .

It is clear that we have T2 = B and that Tis hermitian, positive, compact; we have:

II AfII2 = (AI, Af) = (BI,f) = (T2I,f) = II TfII2

for allfE H 1 , thus

(2.41) II Afll II Tf II, V f E H 1 .

17 For an infinite dimensional Hilbert space; in the finite dimensional case one has u(A) = up(A) as in Theorem 3.


Let

(2.42)

and define U: HI -+ H2 by

(2.43) Ug = Af.

29

We thus have (2.40), U being isometric from HI into H2 from (2.41). The operator U so defined can then be extended, by continuity, to the closure of HI in HI' that is

-- --to say T(H,). The operator U so defined is an isometry from T(H,) into H 2' 0

Remark 4. If N(A) "# {O}, then the closure of HI is not equal to HI' 0

4. Solution of the Equations Au = f. Fredholm Alternative

Let H be a complex Hilbert space «,), I I scalar product and norm in H),

A E 5l'(H) a compact normal operator; (B EJ. the spectral decomposition of H J.eap(A)

with respect to A.

For each f E H, we have f = (!;J.l.Ea.(A) with L Ih 12 < + 00. Then J.eap(A)

Proposition 10. 1° The equation Au .= f admits a solution u if and only if

(2.44) L Ihl2 < + 00 J.eap(A) IAI2

(which implies thatfo = 0 if 0 E (Tp(A)). 2° The solution is unique if and only if (2.45)

3° In the case where 0 E (Tp(A), the general solution is obtained by adjoining to a particular solution an arbitrary element of Eo(A) = kernel of A. 4° the general solution is written in every case as

(2.46) u = L Ii + Uo , Uo E Eo(A) . J.ea(A) - {OJ A

The proof of Proposition 10 is immediate.

Remark 5. Fredholm Alternative. In the statement of Proposition 10, the difficulty arises from the possible inclusion of 0 in (Tp(A) and the convergence of A E (Tp(A) to o. One often encounters a problem where these difficulties are absent. In effect, suppose the equation to be solved in His:

(2.47) (JI,/ - A)u = J, Jl E C\{O} , fE H given.

We then have the following proposition whose proof is also as immediate as that of Proposition 10.


Proposition 11 (Fredholm Alternative). Let H be a complex Hilbert space, A a bounded operator (A E 2(H)) normal and compact, Ji E C - {O},fE H. 10 If Ji ¢ a(A), the equation (2.47) admits a unique solution for each f E H. If f = I h,/;. E E)JA), this solution is given by

AEO'p(A)

(2.48)

2° If Ji E a(A), Ji -# 0, the equation (2.47) only admits a solution if (2.49) 1;,=0;

in this case, the equation (2.47) admits an infinity of solutions obtained by adjoining to one of them an arbitrary element of E/l(A); we have

u = I ~ + U/l' U/l E E/l(A) . AEO'(A)-{/l} Ji - A

(2.50)

The condition for the existence of a solution is

(2.51 ) (f, v) = 0 for all v E E/l(A) = E,,(A*) .

The statement of Proposition 11 calls for the following comment: alternative means that: either the homogeneous equation (the case where f = 0) has only the trivial solution u = 0, which is the case whenever N(JiI - A) = to} [here Ji ¢ a(A)], whence the image of JiI - A is H and the equation (2.47) has a single solution regardless of the second member f E H; or the homogeneous equation has d linearly independent solutions, but then there are solutions for the complete equation if and only if the second member satisfies d linearly independent conditions (f, N (iiI - A *)) = 0; the solution is then indeterminate to order d, that is to say there are d degrees of freedom. Here dimE/l(A) = dimE,,(A*) = d/l(= d). There will thus be d/l degrees offreedom corresponding to the dimension of the space E/l(A). In every case, the number of compatibility conditions given by (2.51) equals the number of degrees of freedom d/l whatever the value of Ji E C - to}. Remark 6. The conclusion of Proposition 11 is not true for Ji = 0 (in effect, the spectrum of A can have 0 as a point of accumulation). The value Ji = 0 thus plays a special role (see Proposition 10). 0

5. Examples of Applications

5.1. Hilbert-Schmidt Operators. Integral Equations

Let E and F be two separable Hilbert spaces. Let A be a continuous linear operator from E to F. We denote by {ediEN an orthonormal basis for E and by {jj LEN an orthonormal basis for F.


00 00

If the series L IIA(ei)II~, L IIA*(jj)lIi are then both convergent18 their sums i=O j=O

are equal and the operator A: E ~ F is said to be a Hilbert-Schmidt operator. The quantity II A II HS defined by

(2.52) def ( )112 ( )112 IIAIIHS = ~ IIA(eJII~ = ~ IIA*(jj)11 2 ,

which is independent ofthe chosen bases in E and F, 19 is called the Hilbert-Schmidt norm of the operator A and we have:

(2.53) IIA IIHs = II A* IIHs . Recall (see Chap. VI, §3) that the Hilbert-Schmidt operators from E to F form a vector space denoted by HS(E; F); this space can be equipped with the positive definite sesquilinear form

(2.54) (A, B)HS = L (A(eJ, B(eJ)F i EN

which is independent of the chosen basis, since the norm associated with this form is the norm II IIHS' Finally, the space HS(E; F) is complete in this norm. It is thus a Hilbert space. Recall that the operators of finite rank are dense in HS(E; F), whence it follows 20

that a Hilbert-Schmidt operator is compact.

Proposition 12. Let: H be a separable complex Hilbert space; A a bounded operator (A E 2'(H)), normal, compact, with spectral decomposition

EB EA (we denote d = dimEA).21 AEt1(A)

Then A is a Hilbert-Schmidt operator if and only if (2.55) L dA\A\2 < +00.

AEt1(A)

We have:

(2.56) ( )112

IIAIIHS = L dAIA\2 AEt1(A)

Note that in practice, we choose an orthonormal basis in each EA; there is then an orthonormal basis {eJiEN in H consisting of eigenvectors ei of A; ei corresponds to the eigenvalue Ai and there are dA; independent eigenvectors corresponding to the eigenvalue Ai'

18 See Gelfand-Vilenkin [I]; moreover the convergence of one of them is sufficient. 19 See Gelfand-Vilenkin [ll 20 In effect, II A II Bs ~ II A II. 21 With the convention Eo = {O} if 0 ¢ O"p(A).


Hence, in this case, counting the eigenvalues according to their multiplicity, we have:

(2.57) IIAIIHS'


We now give an example of Hilbert-Schmidt operators. Let H = L2(1R; dx). Denote by dx dy the Lebesgue measure in 1R2 and let

(2.58)

K is called a kernel.

o

If fE H, then for almost all x E IR, y r+ K(x, y) f(y) is a square-integrable measurable function such that F given by

(2.59) F(x) = L K(x, y)f(y)dy

is a square-integrable measurable function. Let AK be the operator from H into H defined by

(2.60) AK:fr+ F; following (2.59) .

We then have

Proposition 13. The operator AK defined by (2.60) is in .!£ (H) and is a Hilbert-Schmidt operator. The adjoint of AK is At where

def ~--(2.61 ) K*(x, y) = K(y, x) .

In particular AK = At if and only if K(x, y) = K(y, x).

Proof AK E .!£ (H) follows from

r IF(xWdx ~ r IK(x,yWdxdy. r If(yWdy. J~ J~2 J~

To show that AK E HS(H),22 let (wn) be an orthonormal base for L2(1R; dx) which is a separable Hilbert space. A basis of U(1R2; dxdy) can be defined by the functions

where Wm ® wn(x, y) = wm(x). wn(y). It is immediate that the wm.n form an orthonormal system in L2(1R2; dxdy). The closed vector subspace generated by the {wm• n } contains all the functions of the formf ® 9 withf E L2(1R; dx) and 9 E L2(1R; dy) and hence all the linear combina-

def 22 With the notation HS(H) = HS(H, H).


tions of these functions. Now it is known (theory of integration) that these linear combinations are dense in L2(1R2; dxdy) and it follows that the wm,n constitute a hilbert basis for L 2 (1R2; dx d y). Thus K can be expanded with respect to this basis:

K(x, y) = L km,n wm(x)wn(y) , m,n

with

We have

thus

AK(Wn)(X) = L K(x, y)wn(y)dy a.e.,

(AK(wn), wm) = L [ wm(x) L K(x, y)wn(y)dy JdX

= r K(x, y)wm(x)wn(y)dxdy = (K, wm,n)L2(1R2) = km,n J 1R2

To find the adjoint of AK :

(AKJ, g) = L g(x) (L K(x, y)f(y)dy )dX = IL2 K(x, y)f(y)OW dxdy

= L f(y)(L K(x, y)OW dx )dy23 = L f(y) L K(x, y)g(x)dxdy

= L f(x') L K(y', x') g(y')dy' dx' = (J, AK·g) ,

where K*(x, y) = K(y, x). If K. = K*, then AK is hermitian. Conversely, if AK is hermitian, K and K* must define the same operator, whence K - K* is the null operator. Now if K E L2(1R2; dxdy) defines the operator AK = 0 we have

IIAillHS = 0 so that IIKIIL2(1R2) = 0 = K = 0 a.e.,

whence Proposition 13. 0

We can then note the following converse proposition.

23 Applying Fubini's theorem.


Proposition 14. Every Hilbert-Schmidt operator from e(lR; dx) into itself is of the form AK:fH AKf(see (2.60)) where the kernel K satisfies K E e (1R2; dxdy) and is unique.

Proof Let il/t be the mapping K --> AK of e(1R2; dxdy) into HS(H) def

(H = e(lR; dx)). It is linear, continuous and isometric hence injective into its

closed image. The image of il/t contains all the operators of finite rank: this follows since every operator a of finite rank can be written in the form a(f) = L ~i(f)Wi' ~i

1 ~ i ~ n

being a continuous linear form on H and thus identified with an element also denoted by ~i of H, and {Wdi ~ 1 ... n being a basis for the image of a. It follows that

a(f)(x) = L ( ~ ~i(Y)Wi(X) )f(Y)dY ,

thus a is defined by the kernel

n

K(x, y) = L wi(xK(y) . 1

Hence, since the operators of finite rank are dense in HS(H), il/t is a bijection of e(1R2; dx dy) onto HS(H). 0

Remark 7. The operator AK defined by (2.60) is hermitian and positive if and only if K = K* and if

IL2 K(x, y)f(x)f(y)dxdy ~ 0, VfE H. o

Proposition 15. Let K be a kernel satisfying: K E L2(1R2; dxdy) with

K(y, x) = K(x, y) and let AK (with AK E HS(H)) be the hermitian operator in H = L2(1R; dx) corresponding to K. Then there exists an orthonormal basis {wn}nEIll such that Wn is an eigenvector of AK associated with the real eigenvalue An of AK. We have:

i) IIAKllHs = C~N IAnl2 y/2 = (IL2IK(X,yWdXdY y/2;

ii) K(x,y) = L Anwn(x)wn(y) in the sense of e(1R2; dxdy). n EN


Let K E L2(1R2; dxdy), AK E HS(H); the solution of

(l + AK)u = f in H = e(lR;dx)

o

is then given by (J + AK1)f = u where AKI E HS(H) (because if A E HS(H) and if I + A is invertible then its inverse can always be written as I + B with B E HS(H) since (J + A)(J + B) = I + A + B + AB == I, thus A + B + AB = 0 and B = - A - AB is in HS(H)). Hence the integral equation


(2.62) f(x) +L K(x, y)f(y)dy = g(x) a.e.

is solved by:

(2.63) f(x) = L Kl (x, y)g(y)dy + g(x) ,

where K1(x, y) E L2(\R2; dxdy). In particular, if K E L 2(\R2; dx dy), the solution of

(2.64) Af(x) - L K (x, y )f(y) dy = g(x)

will always be under the Fredholm alternative.

Remark 8. Let us give other examples of Hilbert-Schmidt operators. If Q is an open bounded subset of \Rn, then for every integer k > nl2 and all mEN, the injection

H'(f + k(Q) --.. H'(f(Q)

is a Hilbert-Schmidt operator (Adams [IJ, p. 175). Similarly if Q is sufficiently regular, the injection (with k > n12, mEN)

H'n + k(Q) --.. Hm(Q)

is a Hilbert-Schmidt operator. In particular for n = 1, Q = Joc, p[, oc and P finite, the injection Hb(Joc, fJD -+ L2(OC, p) is a Hilbert-Schmidt operator. 0

Remark 9. An operator A in HS(E, F) being compact, we have from Theorem 5 the following property: A has a decomposition of the form

A = UT

where T E HS(E) is a positive Hilbert-Schmidt operator and U is an isometry from

the closure T(E) of the image T(E) of E by T in F. 0

5.2. Nuclear Operators24

Let H be a separable Hilbert space. A Hilbert-Schmidt operator A on H is said to be nuclear (or trace class) if

(2.65)

the An here denoting the eigenvalues of the positive operator T (see (2.40), Theorem 5 and Remark 9), counted according to their multiplicities (see 2.57)). Recall that a positive hermitian operator on the Hilbert space H is an operator of finite trace if for every orthonormal basis of H, {en}neN" the series

00

(2.66) L (Aen, en) < + 00 . n=l

24 See Chap. VI, §2, in the context of Banach spaces.


We note that 00

Tr A = Trace of A = L (Aen, en) ,25 n = 1

which is independent of the chosen basis en (see, for example, Gelfand-Vilenkin [1] ). We have the

Proposition 16. A necessary and sufficient condition for a positive hermitian compact operator to be nuclear is that

(2.67) Tr A < + 00 .

Proof Let A be a positive nuclear operator. Let A 1/2 be defined by:

(2.68) A 1/2 en = jf" en ,

where the An are the eigenvalues26 of A, and {en} IS an orthonormal basis of corresponding eigenvectors. Then A 1/2 E HS(H) since

00 00

L IA1/2enI2 =L An< +00. n = 1 1

Thus for every orthonormal basis of H, Un}

A being hermitian, we have:

00

L (Afn' f,,) 1

thus Tr A < + 00.

Conversely, if A is a posItive hermitian compact operator of finite trace, let {An} n E N*' be the sequence of eigenvalues26a of A, {en} n E N* an orthonormal sequence of corresponding eigenvectors, we then have

00

L (Aen, en) < +00, n = 1

thus A is nuclear. o Proposition 17. i) The product of two Hilbert-Schmidt operators is a nuclear operator. Conversely, every nuclear operator is the product of two Hilbert-Schmidt operators. ii) If A is nuclear, its adjoint A * is nuclear.

25 The scalars (Aen, en) are in effect the diagonal elements of the 'infinite matrix' representing A with respect to the basis en, whence the terminology. 26 Counted here with their multiplicities. 26a Counted here with their multiplicities.


iii) If the operator A is bounded (A E Y(H)) and if the operator B is nuclear, then the operator AB is nuclear.

Proof i) If A and B E HS(H), then AB is compact. Utilising Remark 9, we have: AB = UTwhere T is a positive hermitian compact operator, and U is a unitary operator. If {en}nEN* is an orthonormal basis of eigenvectors of T, corresponding to a sequence of eigenvalues An' we have U(en) = hn:

o ~ An = (Ten, en) = (ABen, hn) = (Ben, A*hn) ,

whence 1

An ~ "2 (IBen12 + IA*hnI2);

if A, BE HS(H) the series LIBenI2,LlA*hnI2 are convergent and we have LAn < + 00; thus AB is nuclear.

Remark 10. The demonstration also holds for AB with:

BE HS(E, F), A = HS(F, G) ;

AB is nuclear from E into G. o Conversely, if A is nuclear, A is compact, thus A UTwith T compact hermitian and positive (or accretive) and nuclear if A is. We have seen that T l /2 is then in HS(H). It is the same for UT l /2 since (U being unitary)

LIUT l /2enl2 = LIT l /2enl2 = LAn; n

thus A = (UTI/2)(TI/2) is indeed the product of two operators in HS(H). ii) If A is nuclear we can write A = (UTI/2)(TI/2), hence A* = TI/2.(UTI/2)*; now (UTI/2)* E HS(H) since UT I/2 E HS(H) and HS(H) contains the adjoints of its members; thus A* is the product of two operators in HS(H). iii) If B is nuclear, we can decompose B = UT with T nuclear and positive (or accretive). If A E Y(H)

AB = (AUT I/2). TI/2 ;

but AUTI/2 is a Hilbert~Schmidt operator, hence AB is nuclear, being the product of two operators in HS(H). 0

The following proposition can then be demonstrated (see for example Gelfand~Vilenkin [1J):

Proposition 18. 1) For the bounded operator A (A E Y(H)) to be nuclear, it is necessary and sufficient that the series L (AIn, gn) shall be convergent for arbitrary

n

systems {In} and {gn} of orthonormal vectors in H. 2) We have

(2.69) n n

where the sup is taken over all orthonormal systems {In}, {gn}.


6. Spectral Decomposition of an Unbounded Self-Adjoint Operator with Compact Inverse

In this section we consider a situation which frequently occurs in the applications of mathematical physics to problems, notably in quantum mechanics. 10 Let H be a complex Hilbert space. Let ( , ) (resp. I I) denote the scalar product (resp. the norm) in H. Let A be a self-adjoint operator, unbounded in H, with domain D(A) dense in H. We equip D(A) with the graph norm luID(A) = (lul 2 + IAuI 2 )1/2, thus making it a Hilbert space, and we make the hypothesis

(2.70) the injection of D(A) into H is compact.

We now prove

Theorem 6. With the preceding hypotheses on A, and if A - I E !t'(H), then

i) IT(A) = ITp(A) c IR. ii) The eigenvalues A E IT p( A) can be arranged as a sequence Pn} n E N such that

IAnl -> + 00 as n -> + 00.

iii) The eigenspaces EA" are finite dimensional and pair-wise orthogonal and

(2.71) H = EB EA"· n EN

Proof 1) First we show that

A - I is compact hermitian.

In effect, let u, v E H; A - I U, A - I V E D( A) and

(A - I U, v) = (A - I u, A (A - I v)) = (A (A - I u), A - I v) = (u, A - IV) ,

thus (A- I)* = A-I. I

On the other hand, A - I is continuous from H into D(A) and hence compact from H into H using (2.70). 2) Now let H = EB Ell be the spectral decomposition of H with respect to

IlE<1p(A - I)

A-I (here 0 if ITp(A- I )). If ull E Ell' vll = A - lUll = J.lUIl and ull = AVIl = J.lA ull ' and since J.l #- 0 we have:

(2.72)

We now show that A has neither a residual nor a continuous spectrum. Since A(D(A)) = H, for aIIfE H, there exists g E D(A) with

g=A-If·

admits a unique solution u E H from the Fredholm alternative for A-I. Since


1 A - 1 u, 1 g E D( A), we deduce that u E D(A). It follows that u is the unique solution

of (,u - A)u = f, r 1 ~ G'p(A-1) ,

thus A E p(A) and G'(A) = G'p(A), whence Theorem 6 follows In the light of Theorem 3. 0

2° We now apply Theorem 6 to a situation very frequently encountered in the rest of this work. Let Vbe another Hilbert space, embedded under a compact injection in the space H, V being dense in H. Denote by I I the norm, and by (( , » the scalar product in V. Let a(u, v) be a sesquilinear form continuous and hermitian on V x Vand coercive on V, that is to say satisfying:

(2.73) { there exists a constant IX > 0 with

a(u, u) ~ IXllull 2 for all u E V.

Let A be a self-adjoint unbounded operator on H with domain D(A) dense in H defined by

{

i) a(u, v) = (Au, v) for all u E D(A) and v E V

(2.74) ii) D(A) = {u E V; V -+ a(u, v) is continuous on V for

the topology on H}

We equip D(A) with the graph norm; D(A) is embedded under continuous injection into V; the injection of V into H being compact, it follows that the injection of D(A) into H is compact. From the Lax-Milgram theorem (see Chap. VI, §3 and Chap. VII, § 1), A is an isomorphism of D(A) onto H and A - 1 E ff'(H), (A - 1)* = A - 1 and A - 1 is compact. One can therefore apply Theorem 6 to A. We then obtain

Theorem 7. Let V c+ H be two Hilbert spaces with compact injection, V being dense in H, a(u, v) a continuous hermitian sesquilinear form on V x V satisfying (2.73); A the self-adjoint unbounded operator with domain D(A) defined by (2.74). Then:

i) G'(A) = G'p(A) = PdkE I'\J with

(2.75) 0 < IX ~ Ak -+ +00 as k -+ +00 ;

ii) the eigenvectors Wk of the operator A normalised in H and associated with the Ak satisfy the variational equation

(2.76) { a(wk' v) = Ak(Wk, v) for all v E V

IWkl = 1 ;


iii) the vector subspace generated by the Wk is dense in V and in H, the Wk forming an orthonormal basis for H.27

Proof From Theorem 6, we immediately have parts i) and ii) and the fact that the Wk form an orthonormal base for H. We show directly that the vector subspace generated by the Wk is dense in V. To do this, we start by noting that a(u, v) defines d E Sf(V) with

(2.77) { a(u, v) = ((du, v)) for all u, v E V

d being an isomorphism of V onto V from (2.73) .

Then let f E V be orthogonal in V to all the Wk and let 9 = d- 1 f E V. We have

0= ((J,Wk)) = ((dg,Wk)) = a(g,wk) = a(wk,g) = Ak(Wk,g)

whence we deduce, Ak being i= 0, that 9 is orthogonal to all the W k in H, and hence 9 = 0 and f = d 9 = O. 0

Remark 11. It will be noted from the properties of continuity and coercivity imposed on a(u, v), that we can take a(u, v) to be a scalar product on V, u --+ (a(u, U))1/2 being a norm equivalent to the initial norm. The Wk thus satisfy, from (2.76),

(2.78) a(wk' wk') = Ak. JZ' (c5Z' Kronecker symbol)

and are orthogonal in V with respect to the form a(u, v), o

7. Sturm-Liouville Problems and Applications

7.1. Sturm-Liouville Problems

Let a, bE IR, and put Q = ]a, b[ and Q = [a, b]; ~m(Q), mEN (resp. L2(Q)) denotes the space (resp. the space of classes) of functions m times continuously differentiable on Q (resp. square integrable in Q) with real values, for the sake of simplicity, We will denote by

II u II 00 = sup lu(x)1 the norm of u in ~O(Q) , XEQ

lui = (J: lu(xWdx )1/2 the norm ofu in L2(Q), We are given

(2,79) {lXk' 13k (k _= 1,2), four real constants, q E ~O(Q) .

A Sturm-Liouville problem associated with (2,79) is a problem which reduces to the following type:

27 Note that the Wk are defined a priori only to within a factor Ak with IAkl = 1. One chooses for iii) a particular Wk for each k. This is implicitly assumed in the sequel, each time one speaks of 'the' orthonormal basis of eigenvectors of the operator A.


Problem (P) given f in L 2 (Q); find u E 'C I (Q) which satisfies:

(2.80) - u" + qu = f in the sense of ~'(Q)

and the boundary conditions:

(2.81 ) { ~Iu(a) + f3lu'(a) = 0 ~2u(b) + f32 u'(b) = 0.28

41

The fact that we are considering only real valued functions here does not make the problem less general: If we choose the operator associated with problem (P) to be self-adjoint, the data (2.79) have necessarily to be real. In this case iff takes complex values, the same holds for u and the real and imaginary parts of u are solutions of a problem (P) with real data. If we now put:

(2.82) {') _ {~k/ 13k for f3k"# 0, k-I Yk - - 1,2

. 0 for 13k = 0 ,

ii) p(u, v) = Y2 u(b)v(b) - YI u(a)v(a), u, v 'E 'C0(.Q)

and

(2.83) a(u, v) = f [ ~: ~: + quv ] dx + p(u, v), u, V E HI (Q) ,29

the Sturm-Liouville problem (P) reduces to the following variational problem (ft), to which it is equivalent:

Problem (ft) find u E V such that

(2.84) a(u, v) = f fv dx for all v E V.

The space V which depends on the boundary conditions (2.81) is a closed subspace of HI (Q) endowed with the norm induced by that of HI (Q); it is defined by

(2.85) V·=

HI(Q) if f3k"# 0 for k = 1 and k = 2,

H 6(Q) if 13k = 0 for k = 1 and k = 2 ,

J-:, = {vEHI(Q);v(a) = O} if 131 = 0; 132"# 0,

Vb = {v E HI(Q); v(b) = O} if 131"# 0; 132 = o.

28 For the same k, Ilk and fJk are not both zero so that (2.81) certainly corresponds to a boundary value problem. 29 The notations a (lower bound of Q) and a = a(u, v) (bilinear form) should cause no confusion.


We will see later that the bilinear form a(u, v) is continuous and V-elliptic under the supplementary condition (2.90) (there exists a constant M > 0 such that q(x) ~ M, \Ix E Q) and that the problem (P) thus admits a solution by the Lax-Milgram theorem (see Chap. VII, § 1).

Proof of the equivalence of (P) and (P). This follows the approach encountered earlier in Chap. VII. First, if u is a solution of (P), then u' E H 1 (Q) by (2.80), and Green's formula (integration by parts here) gives

(2.86) - [u'v]~ + J: (u'v' + quv)dx = J: fvdx for all v E V

whence (2.84) on observing that - [u'v]~ = p(u, v)

holds from (2.81). Thus u is also a solution of (P). Conversely, if u is a solution of (P) then the choice of v E £2'(Q) c V shows that (2.80) is satisfied if (2.84) is. We then deduce that u" E L 2(Q), so that u E H2(Q) ~ re l (Q) and (2.86) follows. Then comparing (2.84) and (2.86) where we take v E V first zero in a then zero in b, we get (2.81). 0

With the goal of establishing the continuity of the bilinear form a(u, v) on the space V x V and its coercivity on V (for suitable q), we shall make use of the following lemma which plainly follows from the properties of the Sobolev space H 1 (Q) (case n = 1) and notably from the compacity of the injection H 1 (Q) ~ reO(Q) (see Chap. IV for the inclusion and for example Adams [1] for the compactness of the inclusion (in dimension 1, this type of result is immediate». For the convenience of the reader, we give here a direct proof of

Lemma 2. F or all e > 0, there exists a constant C, > 0 such that:

(2.87) Ilull~ ~ elu'I2 + C,luI2, \lu E Hl(Q).

Proof i) First proof. It is enough to establish the property for re 1 (Q) which is dense in H 1 (Q). Suppose that (2.87) does not hold. Then there exists e > 0 and a sequence {<Pn}nE N with

(2.88)

such that

(2.89)

{ <Pn E rel~) for all n

II <Pn II 00 - 1

On the one hand, we deduce from (2.89) that I <Pn 12 dx ~ - so that fb 1

a n

mes{ x;l<Pn(x)1 > ~} < ~;


thus we can find (continuity of CPn)xn E Q such that

:!( Jlxn - Ynl.(J: Icp~12dx y/2 :!( ~ Icp~1u ,30

which gives Icp~li2 ~ 1~ -+ + (f) with n. On the other hand, from (2.89) we deduce

that 1 ~ B I cP~ I i2, giving a contradiction. ii) Second proof of Lemma 2 for Q = ]a, b[ (a and b finite). It is enough to establish the property (2.87) for u E ~1 (Q), a space dense in HI (Q). We may always choose a = 0, b = 1. Put

X n + 1

en

(1 - X)" + 1

(1 - e)n e<x:!(l.

The function Oe admits a discontinuity [OJ - 1 at e. Its derivative is given by:

O~(X) =

Xn

(n + 1)-e"

(1 - x)" (n + 1) (1 _ c)"

and using integration by parts we show that:

e<x:!(l,

u(e) = (u', OJ + (u, O~) . Hence, by Cauchy-Schwarz:

lu(e)1 :!( lOci lu'l + IO~ lui· Now

lOci :!( (2n ~ 3 y/2, IO~I:!( (2nn ++ 1)1/2

Whence the result (2.87), on taking n sufficiently large.

30 From the Cauchy-Schwarz inequality.

o


Note that this generalises straight away to Sobolev spaces Wi, P(Q) (see Kato [I] p, 193). We are now in a position to prove

Proposition 19. i) The symmetric bilinear form a(u, v) defined by (2.83) is continuous on V x V endowed with the topology induced by HI (Q) x HI (Q). ii) There exists a constant M > 0 such that if (2.90) q(x) ~ M for all x

then the bilinear form a(u, v) is coercive on V, i.e.

(2.91)

with

(2.92)

{there exists a positive constant IX > 0 such that

a(u, u) ~ IX II U 112 for all u E V

Proof i) Note that

(2.93)

so that (using Cauchy-Schwarz):

la(u,v)1 ~ max(l, IIqlloo)llull.llvll + Ip(u,v)1

~ C II u II II v II , Vu, V E HI (Q)

from (2.93) and Lemma 2, whence part i). ii) From (2.83), we deduce:

a(u,u) = lu'Il2 + I qlul 2 dx + p(u,u) ~ lu'Il2 + I qlul 2 dx - Ip(u,u)l;

from (2.93) and Lemma 2, we have:

Ip(u, u)1 ~ (IYII + IY21Helu'I 2 + CE lu1 2) so that

a(u, u) ~ lu'll2(1 - (IYII + IY21)e) + [ inf-'q(x)1 - CE ] lull2. X E Q

Now let IXE]O, I[ be given; we can choose e> 0 such that 1 - (IYII + IY21)e = IX whence CE is known and the conclusion of Proposition 19 holds for M ~ CE • 0

In the following we suppose that:

(2.94) q E rcO(Q) satisfies (2.90)

which allows us to use the classical theory of ordinary differential equations, but otherwise plays no essential role. Note that the operator A with domain D (A), defined with the aid of the bilinear form a(u, v) by formula (2.74), is given by:


(2.95) {A = - d~2 + q

D(A) = {u E H2(Q); u satisfies (2.81) .31

The injection of V [V c H 1 (Q)] into H = L 2 (Q) being compact, we find ourselves in the situation considered in §2.6: A - 1 is compact and self-adjoint and we can apply to A Theorem 6 of §2.6. In fact, here, we further have: A - 1 is a self-adjoint Hilbert-Schmidt operator32 and hence given by a symmetric kernel K or Green's kernel33 which is determined with the aid of a function G, called the Green's function 34 of the operator A:

Proposition 20. Suppose that A is defined by (2.95) and that (2.90) holds. Then,for all t E Q, there exists a real valued continuousfunction on ti, G,: x 1-+ G,(x) having the following properties: i) in each open interval a < x < t, t < x < b, G, is twice continuously differentiable and satisfies in the sense of distributions in Q:

def (2.96) AG, = (j" «(j, = (j(x - t) Dirac measure at the point t);

ii) G, satisfies the boundary conditions (2.81).

Proof From the elementary theory of linear differential equations, there exists a solution U 1 # 0 (resp. U2 # 0) of the homogeneous equation y" - q(x)y = 0 satisfying the condition

oc 1u1(a) + P1u'l(a) = 0 (resp.oc2 u2 (b) + P2u~(b) = 0);

u1 and U2 are not proportional because otherwise there would exist a solution u =f; 0 of the equation y" - qy = 0 in D(A) which cannot be the case since the kernel of A is {O} if (2.90) holds. Since U1 and U2 are linearly independent, every solution of y" - q(x)y = 0 can be written in a unique manner as y = C1U 1 + C2 U2 where C1 and C2 are constants. Note that the Wronskian W = Ul u~ - U2U'1 of the solutions U1 , U2 is then a constant d # 0 since dW/dx = O. It now suffices to choose the constants C1 and C2

in such a way that the function G, defined by:

(2.97) G,(x) :;: {C1U1(X), C2 U2 (X),

is continuous in t and satisfies:

(2.98) G;(t + 0) - Gat - 0) = -1

so that (2.96) holds.

31 Recall that H2(Q) c; ~I(Q).

32 The space L2(Q) then being taken to be complex. 33 See Chap. VII. 34 See Chap. II and Chap. VII. We have earlier established, in Chap. VII, §3, the existence of such a Green's function. It will be constructed explicitly in the particular case of problem (2.80) with (2.81).


We find we are thus led to the relations:

(2.99)

which consequently give:

whence:

(2.100) Gt(x) =

and Proposition 20 follows.

{C1U1(t) - czuz(t) = 0 c1u'dt) - czu~(t) = 1 ,

U1 (t) . Cz = - -d-'

for all t E Q ,

o It will be noted that for all t E Q, Gt E Hl(Q), but Gt ¢ D(A); AGt = ()t E [Hl(Q)]'. The Green's function Gt satisfies

(2.101)

Now denote by

(2.102)

a(G" v) = v(t) for all v E V.

K: {(X,Y)HK(X,y) = GAy) QxQ-+IR;

K thus defined is a continuous function on Q x Q which can be extended by continuity to y = a, and y = b by putting:

K(x, a) = -uz(x)u1 (a)

K(x, b) = U 1 (x)uz(b)

VXE]a,b[, d

, d

and to x=a, x=b by taking

K(a,y) = -uZ(a)u1 (y)

K(b, y) = U1 (b)uz(y)

VYE[a,b] d

, d

so that, extended in this way, the function K called the Green's kernel of the operator A is of class <;&'0 on Q x Q and satisfies

(2.103) K(x,y) = K(y,x), VX,YEQ.

In particular

(2.104)

K thus defines a Hilbert-Schmidt operator a K

(2.105) (aKf)(x) = J: K(x, y)f(y)dy, x E Q for all fE LZ(Q).


We then have the

Proposition 21. Let A be defined by (2.95) and q satisfy (2.90). Then

(2.106)

in other words, every solution u of problem (P) [or (P)] is given by

(2.107) U(x) = a K(f)(X) = f K (x, y)f(y) dy, f E L 2(0)

and conversely.

Proof Let u be given by (2.107); we have:

U2(X) IX u1(x) fb u(x) = - -d- a u1(y)f(y)dy - -d- x u2(y)f(y)dy.

The derivative in the distribution sense of u is:

u' (x) fX u' (x) fb u'(x) = - T a u1(y)f(y)dy - T x u2(y)f(y)dy

thus U E ~l(Q). Similarly u" is defined by:

U"(X) IX u"(x) fb U"(X) = - T a ul(y)f(y)dy - + x u2(y)f(y)dy

_ [U 1 (x)u~(x) - U'l (x)u2(x)] f(x) d a.e. x EO.

Now

so that

whence (2.80). Now for x = a:

similarly for x = b

u"(x) = + qU(X) - f(x) a.e. x EO,

(J,2u(b) + P2u'(b) = O.

47

Thus (2.81) is satisfied and u is a solution of problem (P) [or (P)]. Conversely, let u be a solution of problem (P) [or (P)]. We have:

a(u, v) = f fv dy for all v E V.


If we take v = Gx which belongs to V, x E Q, we obtain:

a(u, Gx ) = f K(x, y)f(x)dx ;

but a(u, GJ = a(Gx ' u) = u(x) from (2.101), therefore u is given by (2.107). 0

Consider now:

Problem (PA)

find u satisfying:

(2.108) {Au - AU = f, fE L2(Q), A E IR,

plus the boundary conditions (2.81) .

The problem (PA) is equivalent to

Problem (FA) (I - AaK)U = aKf

and for this problem, for whatever A, the Fredholm alternative (see §2.4) is applicable. Denote by An' n E N*, the eigenvalues of A which are real, strictly positive and which can be arranged as an increasing sequence tending to + 00.

Let dn = dim N(A - AnI); then we have

(2.109) " dn L.. A2 < +00.

n E F\j* n

Remark 12. Note that up till now we have assumed that q satisfies (2.90). If this should not be the case, then there exists Ao E IR such that q + Ao ~ M > O. In the preceding treatment, we would replace the operator

d2 _ d 2

A = - dx2 + q by A = - dx2 + q + Ao, q + Ao ~ M .

In problem (P A)' we would recover the operator A by writing d 2 _

A = - dx2 + q + Ao + (A - Ao) = A + (A - Ao) ;

we would thus find that An - Ao > 0, An > Ao, V n ~ 1 and that L 1 ~i 2 < + 00

" dn from L.. IA _ A 2 < + 00.

n 01 o

Remark 13. We have assumed that q is a function continuous on Q.1t follows that the eigenvectors of A are of class ((l2.

Recall that the eigenvectors {Wn}nEN* which form an orthonormal basis of L2(Q) satisfy:

- w: + qWn = An Wn , An # 0 ,

thus Wn is of class ((l2 if q is of class ((l0, and of class ((lm + 2 if q is of class ((lm. We deduce that if q is of class ((lao, then the Wn are also of class ((l0C!. 0


Remark 14. A (useful) variant of Problem (P l) is obtained in the following manner. Let p be a given function satisfying:

(2.110)

We replace equation (2.108) of Problem (P l) by

(2.111) Au - PAu = f (or pf),

the boundary conditions being otherwise unchanged. 35 We then introduce the space L~(Q) (space L 2 with weights equal to p), that is to say the space of

measurable functions g with values in IR (or q such that f !g(xW p(x) dx < + 00.

Equipped with the norm (f !g(xW p(x)dx y12, L;(Q) is a Hilbert space and we

have:

(2.112)

For the problem of the eigenvalues associated with (2.111) we will then obtain the eigenfunctions which form an orthonormal basis for L~(Q). 0

Remark 15. More generally, we can consider the same type of problems for the operator

(2.113)

where we have:

(2.114)

with

(2.115)

L = - ~ [k(X) ~ ] + q/ d~ dx

{i) k > 0 on Q

or

ii) k > 0 In Q, k ~ 0 on Q.

The problem (P), (P) or (P l) associated with the hypothesis (2.115) i) are the "non degenerate" Sturm-Liouville problems. They reduce to the problems of type (P) by changing the variable and the function: for example, if we put:

_ 1 _ IX dO' _ 1/2 (2.116) v- k1/4 u, t- o K(O') , K-k

the problem of eigenvalues with boundary conditions (2.81) on Q:

Lu = AU

35 We note that the traces ofzero order (i.e. here u(a) and u(b» and of order I on r = oQ = {a} u {b}

{ du d 2u } of the u E W = U E L;(Q); -, - E L2(Q) exist, which allows us to formulate (2.81).

dx dx 2


is transformed into an analogous problem on Q:

( Q = J~, P[ ~ = f: :(:) , p = f: :~J for the operator L such that

(2.117)

where ij is defined by:

d2 X

(2.118) ij(t) = - - (x) + q(x) , dx2 I X x=x(l)

X(x) = I k(x)1 1/4 ,

which assumes k E ~2 (.0) in order that ij shall satisfy (2.79). For a problem which introduces an equation analogous to (2.111) with A replaced by L we put:

(2.119) v = i, X = [pkr/4, t = J: J f (a)da ;

v then satisfies

d2X

(2.120) Lv = AV, ij(t) = - dx2 (x) + ~ (x) I

X P x=x(l)

Remark 16. Degenerate problems. These are relative to the case:

d2

A = - dx 2 + q, q E ct'(Q),

(2.121)

L = - ~ (k~) + q, dx dx

q unbounded in .0

k satisfying (2.115)ii) ,36

q bounded or not in Q ,

o

these operators glVlng rise to differential equations which degenerate at the boundary of the open set. We shall give examples (Legendre and Chebyshev operators) later. In variational form, this type of problem generally involves spaces analogous to Sobolev spaces, but with "weights". We can consider boundary conditions analogous to (2.81) only if the function spaces involved possess trace properties (of order 0 and 1) on the boundary. In general, the boundary conditions are part of the features of a space (see below 7.3 Legendre operator, 7.6 Chebyshev operator). An analogous situation is encountered when we replace the open set Q by an unbounded interval in ~ (see below 7.4 and 7.5 Hermite operator, Laguerre operator, ... ). 0

36 And k vanishes at at least one point of the boundary.


Remark 17. Most of the traditional orthonormal bases are derived from a Hilbert-Schmidt kernel inverting a 2nd-order differential operator or more generally from the situation encountered in §2.6. With some additional theoretical difficulties, the theory extends to partial differential equations in [W: so as, for example, if Q is a bounded open set in [Rn with boundary r sufficiently regular, to make it possible to solve

(2.122) {-L1U=f,

with boundary conditions on r = a Q .

Again we find in this type of problem a Green's kernel K which is no longer a function continuous on Q x Q, nor in the space

L2(Q x Q,dx.dy) , dx = dx 1 ... dxn , dy = dYl ... dYn.

Nevertheless, this kernel K defines an hermitian operator a K' which though no longer Hilbert-Schmidt, remains compact, thus allowing the application of the results from §2.6. 37 0

We now show how some known bases can be obtained by applications of the preceding theory.

7.2. Trigonometric Series or Bases38

i) First we consider the problem:

(2.123) {

- ~2~ = f, ° ~ x ~ 1

u(o) : u(1) = ° . This is a particular case of problem (P) with a = 0, b = 1, q = 0, 13k = 0, k = 1, 2. The problem (2.123) corresponds to the Dirichlet problem for the

d2

operator A = - dx 2 •

The eigenvalues and eigenvectors of the operator A satisfy:

(2.124)

q E rc oo (since q is defined by:

(2.125)

{ An = n2 n2, n;?; 1 and n E N

wn(x) = j2 sin nnx ;

= 0) and thus Wn E rcro(Q).39 We note that the Green's function Gt

G,(x) {(I - t)x,

t(1 - x),

37 See Chaps. II and VII for examples. See also Remark 8 which implies that for m sufficiently large, the Green's kernel Km of (_LJ)m defines a Hilbert-Schmidt operator a K _.

38 See Fourier series in Chap. IlIA, § I and III B, §6. 39 See the relevant result given in Remark 13.


and the kernel K by:

(2.126) K(x, y) = {(1 - x)y, 0 < y < x x(l - y), x < y < 1 ;

the Wk form an orthonormal basis for U(O, 1). ii) Next we consider the problem

(2.127) {

dZu - dxz + U= f, 0 < x < 1

u'(O) = u'(l) = 0 .

This is again a particular case of problem (P) with a = 0, b = 1, q = 1, 0(1 = O(z = O.

dZ The problem (2.127) is the Neumann problem for the operator A = - dxz + 1.

The eigenvalues and eigenvectors of A are found to be:

(2.128) {

An = 1 + nZn Z, . n EN,

Wo(x) = 1 ,

wn(x) = .j2 cos nnx, n ~ 1 .

The Green's function Gt is obtained explicitly as follows: the solution (to within a multiplicative constant) U 1 of Au = 0 satisfying U'1 (0) = 0 is u1(x) = cosh(x); a solution Uz of Au = 0 satisfying u~(l) = 0 is

uz(x) = [eX + e-(X-Z)].

Then Wronskian W = UIU~ - UZU'1 = d = 1 - e Z, so that Gt is defined by:

{ e2 ~ 1 .[e t + e-(t-Z)].cosh(x), 0 < x < t

Gt(x) = 1

2 . cosh (t). [eX + e-(x-2)], t < x < 1. e - 1

(2.129)

The W k which are '1]00 (q = 1) again form an orthonormal basis for LZ(O, 1). iii) Since every functionfin LZ( -1, + 1) is the sum of an even functionk and an odd function fo,

1'( ) = f(x) + f(-x) JE X 2 '

I' ( ) = f(x) - f( -x) JO X 2 '

the expansions

(2.130) { k(x) = 0(0 + L O(n cos nnx ;

n ~ 1

fo(X) = L Pn sin nnx ; n ~ 1

L1O(.I Z < + 00

LIP.I Z < + ex)


which are valid in the L2(0, 1) sense are again valid in U( -1, + 1) and give:

(2.131 ) { f(x) = lXo + I (IXn cos nnx + Pn sin nnx) = I Yn e - im,x

n;;'1 nEZ

I (1 1Xn1 2 + IPnI 2 ) < + CIJ , n?: 1

valid in L2(-1, +1).

This proves that {e- im,xlj2}nE.I' is an orthonormal basis for the space L2( -1, + 1),40 or again that {e- in6Ifo}nE.I' is an orthonormal basis for the space L 2(r), where r = {z = ei6; ° ::s: () ::s: 2n} is the unit circle in the complex plane.

7.3. Legendre Operators

We now give an example of a degenerate problem on an open bounded set in IR. Let Q = ] -1, + 1 [, and consider:

(2.132)

2m will be called the Legendre operator 01 index m. This operator belongs to the class discussed in Remark 16 with

(2.133) m2 { k(x) = (l - x 2), k E ~1(i2}, kliJQ = 0, k'liJQ '# ° q(x) = 1 _ x 2 ,q E ~O(Q), q unbounded in Q for m '# 0.

The domain of these operators will be examined below (see Proposition 23).

7.3.1. Study of the case m=O. Legendre polynomials. We introduce the space

(2.134) V = {u E L2(Q); )1 - x 2 :: E L 2 (Q)} and define, for all u, v E V, a(u, v) by

(2.135) a(u, v) = f:: [(1 - x 2 ) :::: + uv JdX; a(u, v) is a scalar product on V and the norm II u II = ) a(u, u) makes Va Hilbert space. For given 1 in L 2(Q) = H, there exists by the Lax-Milgram theorem a unique u E V such that

(2.136) a(u, v) = L1vdX for all v E V.

40 This has been shown by another method in Chap. IlIA, § l.


Remark 18. 1 ° Here we have H 1 (Q) <:+ V. 2° If u E V, u is in H foc(Q), hence in <t'0(Q), but u 1: <t'0(Q) so that one cannot in general define the trace of u E Von r = {-I} u { + 1 } (i.e. the values of u( -1) and u( + 1». We show this by an example. After localising and changing the variable it will be enough to verify it (for example) in the space

(2.137) V1 = {u E U(O, 1); jtu' E U(O, I)} ;

Since it 1: L 2 (0, 1), we can find 9 ~ 0, gEL 2 (Q) such that

(2.138) Il g~ dt = + 00 (e.g. taking g(t) = Jr - 1 ). ° V t t Logt

Then putting

(2.139) u(t) = - - du , Il g(u)

,Ja u cannot be bounded in a neighbourhood of t = 0. By Cauchy-Schwarz we have:

I lu(tWdt:( I [LOg~ r Ig(uWdu Jdt:( Il9(tWdt

whilst

u'(t) = g(t) jt

a.e.

thus u E V1.

We now demonstrate

o

Proposition 22. Let H = L2(Q) with Q = ] -I, + 1 [ and V defined by (2.134). Then

i) <t' 1 (Q) is dense in V, ii) the injection of V into H is compact.

iii) If V = {u E ~'(Q); Jl - x 2 u E L2(Q), Jl - x 2 u' E L 2 (Q)},

(2.140) V = V (equivalent norms) .

Proof i) First we note that if v E V then w = (1 - x 2 )v is in H 1 (Q) and since n = 1, w E <t'0(Q). But then w( + 1) = w( -1) = 0, so that w E H 6(Q) because

otherwise v would be equivalent to 2 C in a neighbourhood of + 1 or - 1 and x-I

would not be in L2(Q). Having noted this, let Vo E Vbe orthogonal to <t'l(Q) in V, then Vo is orthogonal to the polynomials in V.


If un = polynomial of degree exactly n, then since (1 - x2 )vo -+ 0 as x -+ + 1 or x -+ -1, we have:

(2.141) o = a(un, vo) = f~: (dun)vodx

where dUn = (2"0 + l)un is a polynomial of degree exactly equal to deg Un = n. Every polynomial of degree n is then a linear combination of the {du k } where deguk = k ~ n. Hence from (2.141) Vo is orthogonal in L2(Q) to all polynomials, and thus to IC O(Q) (Weierstrass' theorem) and by density to L 2(Q). Thus Vo = 0 and part i) then follows. ii) Now let {uv } V E F\J be a sequence weakly convergent to U in V, II Uv II ~ C, '<Iv E N. For sufficiently small fixed positive s, introduce q> E ~(Q) such that

(2.142) {suppq> c [- 1 + s,1 - s], q>(x) E [0, 1]

q>(x) = 1, x E [- 1 + 2s,1 - 2s] .

Since {q>u v } is bounded in H 1 (Q) and the latter is contained with compact injection in L2(Q), we can further suppose that Uv is such that q>u v -+ q>u strongly in L2(Q). Then we have:

(2.143)

Now observe that

(2.144) { r Iw(xWdx ~ 2slwli2(Q) + IjJ(s)IJI Jl- 2s":: Ixl":: 1 where IjJ(s) -+ 0 with s, for all w E V.

2 '12 - X W P(O)

In effect, (2.144) follows from the inequality, valid for w E 1C 1 (Q),

(2.145) Iw(xW ~ 2lw(tW + 21 LOg (1 + t)(1 - X)I fX Jl _ (J21~wI2d(J (1 + x)(1 - t) r u(J

by first integrating with respect to t on ] - 1, + 1 [, then with respect to x on ] -1 + 2s, 1 [. Since II UV - U II ~ constant, we can choose s small enough to ensure that

(2.146) 2 r Iuv - ul 2dx ~ t: (t: > 0 given) Jl- 2s":: Ixl":: 1 With this choice of s, (2.143) shows that f~: luv - ul 2 dx -+ 0 as v -+ +00,

whence part ii).


iii) It is immediate that V <:+ V. That V <:+ V, will follow if we can show that

(2.147) .y = {VE~'(IR+); jtvEL2(1R+), jtV'EU(IR+)} <:+ L 2(1R+).41

This space is studied in greater detail in Sect. 7.5 below (the Laguerre operator) where it is shown that ~(IR +) is dense in .y. It follows from the proof of this fact that we can restrict ourselves to proving (2.147) by considering only functions v E .y of class CC 1 which are null for large enough t. We then have:

f +OO d f+oo Iv(tW = - t dO' Iv(O')1 2 dO' = - 2 t v(O')v'(o') dO'

whence

L+ 00 Iv(tW dt ~ 2 L+ 00 dt 1+ 00 Iv(O')llv'(O')1 dO' = 2 f: 00 O'lv(O')llv'(O')1 dO'

~ 21fo VIL'(G;l+)lfo v'IL2(G;l+)

whence part iii).

Corollary 1 (to Proposition 22). ~(Q) is dense in V.

o

Starting from the fact that CC 1 (Q) is dense in V, a truncation followed by a regularisation proves the result [see the proof of Lemma 6 treating the case of Laguerre operators]. 0

We now consider the operator d = !Eo + I with domain D(d) = {u E V; V -+ a(u, v) is continuous on V in the U(Q) topology}; it is self-adjoint42 with compact inverse; we can thus apply the theory of §2.6. Before making precise the properties of the eigenvalues and eigenvectors of d, we can characterise D(d) by proving

Proposition 23. i) The operator d is an isomorphism of

(2.148)

onto L 2(Q). ii) More generally, the operator d is an isomorphism of

(2.149) Dk(d) = {uEH k+1 (Q),(1 - X2)UEHk+2(Q)} ontoHk(Q),kEN*.

Proof a) We start by making use of a lemma related to some of Hardy's ineq uali ties. Observe that if <p is a sufficiently regular function, then we have

(2.150) 1 it d <p(t) = - -d (O'<p(O'))dO' toO'

41 RecallthatlR+ = ]0, +oo[,lR+ = [0, +00[. 42 The space L2(Q) then being taken to be complex.


and, generally,

(2.151)

we then have

dk + 1

Lemma 3 (Hardy's Lemma). If for kEN, "I<+l (t<p(t» E L 2(0, + (0) (resp. dt

L2(0, T), T > 0), then

(2.152)

and then:

(2.153)

dk + 1

Proof of Lemma. Put tJ;(t) = dt k + 1 (t<p}(t); we have:

Iddk:1 = I k~l [t aktJ;(a)dal = If1 ~ktJ;(t~)d~1 ~ f1 ~kltJ;(t~)IL2d~ t L 2 t Jo L 2 0 L 2 0

f 1 ~k (f+OO )1/2 = o.}Z 0 1 tJ;(xW dx d~

and the lemma follows on using (2.151) and noting that

f 1 ~k - 1/2 d~ = 2 . o 2k + 1

o

b) Part i) of the Proposition. We see immediately that if u E H 1 (Q), (1 - x 2)U E H 2(Q), then (integrating by parts)

a(u,v) = - [ ~[(l- X2 )dUJVdX forall VE~l(Q), JQ dx dx

hence:

thus U E D(.s;f). Conversely:

(2.154)

la(u, v)1 ~ c(u)lvlu(Q)'

{

U E V

U E D(d) = d [ - - (1 dx


The conditions: u E D(d) and u E HI(Q) imply (1 - x 2)u E H2(Q). In effect, we then have:

d 2 2 ,d 2 du 2 dx2 ((1 - x )u) = - 2u - 2xu + dx (1 - x ) dx E L (Q).

From (2.154), to establish (2.148), it then remains to show that dujdx E L 2(Q); but this follows from djdx[(1 - x 2)dujdx] E L2(Q) and from Hardy's lemma for k = 0 by localisation and change of variable. Thus we have (2.148) and that the norms are equivalent, whence part i).

c) Part ii). Part ii) is now established by making repeated use of Hardy's lemma. 0

Remark 19. It follows from Proposition 23 that we have

00

(2.155) n Dk(d) = ~oo(Q) . k=O

Consequently, if in (2.136),f is given in ~ 00 (Q) then so also is the solution u. The operator d is therefore hypo-eJliptic.43

Spectrum of the operator 2 0 ,

The eigenfunctions of the operator 20 are the same as those of the operator d. It then follows from Proposition 23 and Remark 19 that the eigenfunctions are in ~ 00 (Q). Thus we find ourselves reduced to studying the classical solutions of the second order linear differential equation

(2.156) (1 - :x2)y" - 2xy' + 2y = 0

which are simultaneously regular at + 1 and -1. The classical theory invites the study of two linearly independent solutions of (2.156) expressed as entire series of the form L a.x'.

, By substituting these in (2.156) we obtain a two-term recurrence relation for the an:

(2.157) (n + 2)(n + l)an +2 = a,[n(n + 1) - 2], n ~ 0

which shows that if 2 #- p(p + 1), pEN, then there are an even series and an odd series which are solutions of (2.156) with radius of convergence equal to 1, but which are unbounded on the boundary of Q, and thus not acceptable. If 2 = (n + l)n, one solution of (2.156) is a polynomial Po. For n = 0, Po is a multiple of 1; a solution linearly independent of Po is then

1 1 + x Qo(x) = "2 Log 1 _ x for Ixl < 1 .

43 See the definition in Chap. V, §2.1.4. Here, with the regularity on the boundary, we have a somewhat stronger property.


Hence for A = 0, the general solution of (2.156) is, for I t I < 1

(2.158) y (x) = APo (x) + BQo(x) , A and B arbitrary constants.

Now Qo which is in L 2(Q) is not in V so that the only acceptable solution is y(x) = APo = constant.44 The general solution of (2.156) for Ixl < 1 for A = n(n + 1), is given, successively, by:

y(x) = APn(x) + B ITn - 1 (x) + -2- Log -1 -(2.159) - x {

[ Pn(x) 1 + x] lIn _ 1 (X) = a well determined polynomial of degree ~ n - 1 .

We should take B = 0 because of the presence of the logarithmic term so that we then obtain:

(2.160) it is simple .44a {

the spectrum of se 0 consists of the An = n(n + 1), n EN;

The corresponding eigenfunctions are polynomials of degree n .

The polynomials Pn , solutions of (2.156) for An = n(n + 1), are then called the Legendre polynomials; they satisfy:

(2.161) f:l1 P;(x)dx = 1 .

We now give the properties of the Legendre polynomials, leaving the reader to verify the given formulae. To start with, in order to calculate explicitly the polynomials Pn to within a multiplicative constant, we can make use of the recurrence relation in (2.157) which shows that each Pn is of degree n and has the same parity as n. A method which is both interesting and which gives all the properties quite simply is the following one which is related to the Laplace transform: C? being a closed contour encircling the real segment [-1, + IJ in the complex plane, we put

(2.162) Jc? {

y(x) = r (x - 0")" f(O") dO"

f and ex determined so that y shall be a solution of (2.156) for Ixl < 1 and A = n(n + 1).

This leads to the differential equation for f:

(2.163) d2 d

d0"2 [(1 - 0"2)fJ - 2ex dO" (O"f) + [n(n + 1) - ex(ex + I)Jf = 0

an equation which simplifies for ex = - (n + 1).

44 Qo can also be eliminated on remarking that Qo is not regular for Ixl = 1. 44. I.e. each An has multiplicity 1. See also further, Chap. 8, §3.3.2, p. 160.


A particular solution of this equation is then

(2.164) f«(1) = Kn«(12 - l)n, Kn being an arbitrary constant,

which then leads to

(2.165) i «(12 - 1)n y(x) = Kn ) _ 1 d(1 . (x - (1 n

C)'

The second term of (2.165) is manifestly a polynomial of degree n, a solution of (2.156) with A = n(n + 1), and therefore is proportional to the Legendre polynomial of degree n.

The coefficient of proportionality being such that t P;(x) dx = 1, we obtain

(2.166) _ (2n + 1 )1/2 Iii «(12 - l)n

Pn(X) - 2 2n 2ni «(1 _ x)" + 1 d(1 C)'

(the SchHifli formula), where C /' is the unit circle oriented in the direct sense, and the Cauchy formulas for holomorphic functions give us

(2.167) Pn(x) = - - - [(x 2 - 1)"] . ( 2n + 1 )1/2 1 1 dn

2 2n n! dxn

Using (2.167) we can easily verify that the Legendre polynomials are orthonormal on] -1, + 1 [ and that they form an orthonormal basis for L 2(Q) by utilising the Weierstrass theorem. The theory that we have developed further gives us:

a(Pn, Pm) = An t PnPmdx = An<5:,

and

(2.168)

If we put

(2.169)

we get the recurrence formulas:

(2.170)

(2.171)

(n + I)Pn + 1 - (2n + l)xPn + nPn - 1 = 0

nPn = xP~ - P~ - l' nPn - 1 = P~ - XP~ - 1 .

Furthermore, using (2.166) we can note that (for all A E IR):

Anp(x) =_1 i (~)n «(12_1)n d(1 n 2in 2 «(1 _ x)n + 1 '

C)'


so that as a describes C?, we have:

la - xl > 1 - Ixl

hence

I(A)"(a2 - 1)"1 IAI" "2 (a - x)" < (1 - Ixl)" <

if I A I < 1 - I x I .

We can then integrate term by term on C? the geometric series

(A)" (a2 - 1)" L"2 (a _ x)" + 1

and obtain

(2.172) + 00 - 1 i da L A"P"(X) = -.

"=0 2m C/'a-x-~(a2-1)

whence, from the residue theorem, we deduce that

(2.173) + 00 _ 1 L A"P (x) = -,=====:c

.n = 0 n ) 1 - 2Ax + A 2

The function of A, x given on the right-hand-side of (2.173) is called the generating function for Legendre polynomials. Starting from (2.173) we deduce

(2.174) Pn(l) = 1 for all n E N

and we can note that

(2.175) xe[-1, + 11

Finally, the zeros of each Legendre polynomial Pn are real and distinct, there are n of them, and they belong to Q.

7.3.2. Study ofthe Case m # 0, mE lL. Legendre Functions. It suffices to consider the operator

for m E N*. With !i'm we naturally associate the symmetric bilinear form am(u, v) defined by

(2.176) am(u, v) = fn [(1 - x 2 ) du dv + m2 2 uv -JdX , Jl, dx dx 1 - x

and the space

(2.177) "f/' = {v E .@'(Q); -,jr=I=~=X=;:2 E L2(Q) , )1 - x2 :~ E U(Q)}


which is a Hilbert space for the norm

(2.178) ]V]2 ] )1/2

+ 1 2 dx ; -x

and we then have

Proposition 24. Let H be the Hilbert space L2(Q) (with Q = ] -1, + 1[) and i/ the space defined by (2.177). Let

Then

(2.179) i/ =..f with equivalence of norms .

Proof This reduces to the following situation: Let:

Q = ]0, a[a < + 00 arbitrary.

(2.180) i/1 = {v E .9&'(Q); Jx E L2(Q), Jxv' E L 2(Q)} ,

..f1 = {v E .9&'(Q); v E L2(Q), Jxv E H1(Q)};

we have to show that i/1 = ill. Let v E i/ 1 then:

whence i/ 1 ~ ..fl. A v-

Conversely, if v E i/1, it suffices to see that Jx E L2(Q).

r:: - d r:: d ( v) -Now (y xv) E H1 (Q). Hence dx (y x. v) = dx x. Jx E L2(Q), and from

v v-Lemma 3 (Hardy's lemma) (applied with cp = Jx and k = 0, Jx E L 2 (Q», we

immediately deduce that Jxv' E L2(Q) whence ill ~ i/1 and the result ~~ 0

Remark 20. We have:

(2.181) lim (J1 - x2 v(x» = lim (~v(x» = 0, v E i/ = i/ . x-+l x--l

In effect, Jl - x2 v E H1(Q) ~ CCO(Q); then the limits of J1 - x2 v as x -+ + 1 or x -+ -1 exists and if they are different from 0, then v rf; L2(Q). 0


Remark 21. We have: 11 ~ V (V defined by (2.134» such that the injection 11 ~ II = L2(Q) is compact and §2.6 applies. Then for the operator!i'm we can develop an account of its properties analogous to those developed for the operator !i' 0 + I. We content ourselves with the following remarks. If we take as our unknown function v defined by

(2.182) u(x) = (1 - x 2 t l2 v(x)

in the equation

(2.183)

we obtain

(2.184) (1 - X2)V" - 2(m + l)xv' + (A. - m(m + 1»v = 0 .

On the other hand, if we differentiate the differential equation (2.156) m times we get precisely (2.184) for v = y<m). We then readily deduce the spectrum of the operator !i'm: put

(2.185) dm _

P':(x) = (- l)m(1 - x 2 )m12 dxm Pn(x) m, n EN,

and note that

(2.186) P':(x) = 0 for m > n ,

then the sequence of functions defined by

(2.187) m _ (2n + 1 )1/2 [(n - m)! J1 /2 -m Pn(X) - 2 (n + m)! p.(x), n = m, m + 1, ...

satisfies

(2.188) !i'mP': = n(n + l)P':, n = m, m + 1, ...

and these are the only eigenfunctions45 of !i'm, mEN. Thus they form an orthonormal basis of U(Q) (and of 11). The functions P::' (resp. P::'» are called the Legendre functions associated with the Legendre polynomials p. (resp. p"). 0

Remark 22. Operators of Legendre type can be generalised by replacing the open interval Q = ] -1, + 1 [ by a regular bounded open subset in IR". For example, let Q be a bounded open subset of IR" such that Q is a bounded manifold of class rt5 oo •

We suppose that there is real-valued function cp of IR' into IR of class rt5 00 such that:

(2.189) {

Q = {x E IR"; cp(x) > O}

r = {x E IR'; cp(x) = O}

dcp(x) i= 0 for x E r;

45 To within multiplicative coefficients.


we can then consider the degenerate differential operator

(2.190)

which is an operator analogous to the Legendre operator 2'0' For a study of this kind of operator and other developments, one can consult Baouendi [lJ and Baouendi-Goulaouic [1] and [2]. 0

7.4. Harmonic Oscillator and Hermite Polynomials

Let Jf = ~ A = ~ [- d 22 + X2 J be the hamiltonian of the harmonic oscillator

2 2 dx of dimension n = 1. If u, cp E !0(IR), then, < ) denoting the duality between !0'(IR) and !0(IR), we have:

(2.191) <Au, cp) = f~ [:: ~~ + x 2 uq> JdX ~ a(u, cp) ,46

an expression which retains its meaning for u, cp E V where V is defined as the completion of !0(IR) in the norm

(2.192) II u II = [a(u, U)]1/2 ;

V is a Hilbert space under this norm. We now prove

Proposition 25. Let H = L2(1R); then

(2.193) V '+ H with compact injection .

Proof i) First we note that for cP E !0(IR), we can find to E IR (such that cp(to) = 0 and thus) such that

fx dcp cp(x) = -d (a)da

to a so that for p >

(2.194)

46 We use complex spaces in this section because the harmonic oscillator is studied in quantum mechanics in such a setting (see Chap. IA,.§6). Nonetheless, we can restrict ourselves to real spaces if we are interested only in Hermite polynomials.


hence we have:

(2.195) { 3 C > 0, such that

IIPI = (L IIP(xWdx yl2 ~ CliIPII, VIP E i?C(IR) ,

and also

(2.196) { 3Cl > 0, such that

IIPIHI(~) ~ CllilPll, for all IP E i?C(IR).

From (2.195) we deduce that the mapping IP f-+ IP of i?C(IR) endowed with the norm in (2.192) into i?C(IR) given the topology of H is continuous. By the density of i?C(IR) in V, this mapping which is injective is extended to a continuous injection of V into H.47

In an analogous way, we deduce from (2.196) that the injection of Vinto Hl(lR) is continuous. ii) We now show that the injection of V into H is compact.48

Let IPEi?C(IR), with 0 ~ IP ~ 1, IP(x) = 1 for XE[-p, +p] with supp IP c ] - 2p, + 2p [, p > 0 given and fixed. Let {un}nE N' Un E V, be a sequence with

(2.197) II Un II ~ C.

Then I Un IHI (Dl) is also bounded; we can extract from the Un a subsequence Uv which converges weakly to u in V and Hl (IR). Then Vv = IPU v converges weakly to v = IPU in H!i(] -2p, +2p[), hence strongly to v in L2( -2p, +2p). Now we have:

(2.198)

r Iuv - uI 2dx ~ 2[ f+ 2p Ivv - vl2dx + i (1 - IP)2luv - U I2 dX] J~ - 2p Ixl ;;, p

and

[with C' = 4C2, C = the constant appearing in (2.197)]. Taking E > 0 given, we can choose p such that

(2.200)

47 For this we need to show that if Un E !0 (IR), Un -+ U in V, and Un -+ 0 in H, then U = 0; the proof uses integration by parts and the Cauchy-Schwarz inequality. 48 We note that V c: H'(IR), but that the injection of H'(IR) into L2(1R) is not compact (IR is not bounded).


from (2.198)-(2.199), we deduce that

(2.201)

Since f+ 2p ivv - vi 2 dx ...... 0 as v ...... 00, p fixed, Uv ...... u strongly in H; whence - 2p

Proposition 25. 0

Moreover, Vis plainly dense in Hand a(u, v) is coercive on V so that we can again apply here the preceding theory. The study of the eigenfunctions of the operator A leads to the examination of the ordinary differential equation:

(2.202) y" + (A - x 2 )y = 0 for A ~ 1 .

By making the following change of the unknown function in (2.202):

(2.203)

we note that z satisfies

(2.204) z" - 2xz' + (A - 1)z = 0 .

Equation (2.204) admits no finite singularity, and if we study the solutions in the

form of an entire series L a.x·, we find, for the determination of the coefficients n ~ 0

an, the recurrence relation

(2.205) (n + 2)(n + l)an + 2 = an [2n + 1 - A], n ~ o. For A -:P 2q + 1, q E N, equation (2.205) admits an even solution and an odd solution (which are thus linearly independent), both expandable in an entire series throughout the complex plane; this is not acceptable because the singularity at infinity is too strong and e - x 2 j2 Z ¢: U (IR) if z is such a solution. Whenever A = 2q + 1, q E N one of the previous solutions is a polynomial of degree exactly equal to q; this is acceptable because, if Wq is such a polynomial solution of (2.204), e- x 2 j2 Wq is square integrable on the right. In summary, we have established

Proposition 26. The spectrum of the operator A consists of the real numbers

(2.206) An = 2n + 1 n EN;

the eigenfunctions of A, called Hermite functions, are of the form 49

(2.207)

where the Wn are polynomials of degree n, called Hermite polynomials, and such that:

(2.208) f:: cP.cPmdx = 15;:', n, mEN.

49 To within a multiplicative constant.


The qJn constitute an orthonormal basis of H = L 2 (IR), and an orthogonal basis of V. To determine explicitly the Hermite polynomials, we can use the recurrence relation (2.205), but we can also utilise a variant of the Laplace transform which will give us the Wn (to within a multiplicative constant) in an interesting form. In effect, we put

(2.209) Wn(X) = -21 . rex' fPn{-r) d, , 1t1 Jel'

where C /' is a path in the complex plane to be determined and fPn a function such that (2.209) defines a solution of (2.204) with An = 2n + 1; fPn must formally satisfy

we then easily find that the function fPn defined by

(2.211) {_I _ ,2/4

fP n ( ,) - kn ,n + 1 e

kn = constant, n E N

satisfies (2.210) on taking C /' to be a closed path oriented in the direct sense. With (2.209), we can deduce the following form for the Wn:

(2.212)

, e- (x -"2)2

+ 1 d" ,n

where we see that wn(x) is different from zero if the closed contour C /' is chosen to encircle the origin and wn(x) then appears as the residue at the origin of the integral of the second term in (2.212); thus we obtain:

(2.213) k dn

( ) _ ( 1 )n n x 2 - x 2 W X - - --e-e

n 2n. n! dxn '

the constant kn being determined on writing

(2.214) f:: e- x 2 w;(x)dx = 1 ,

which gives

(2.215) 2n. n!

In The Hermite functions are thus defined by:

(2.216)


In the scientific literature, the name 'Hermite polynomials' is generally given to the polynomials Pn defined by

(2.217)

We then have the following formulas:

(2.218)

(x, t) H e - 12 + 21x is called the generating function for the Hermite polynomials;

(2.219)

(2.220)

Pn + 1 - 2xPn + 2nPn - 1 = 0, n ~ 1

P~ = 2nPn _ 1, n ~ 1 .

Let us finally note that the functions (/>n defined by (2.216) belong to the space 9' (IR) of functions rapidly decreasing on IR. Denoting by :F the Fourier transform defined on 9'(IR) by fH j,

~ 1 i . f(y) = h:: e- 1xy f(x)dx , v' 2n ~

it is easy to verify that the operator A and:F commute. We know that :F extends to L2(1R) as a unitary transformation of L2(1R) onto itself. 50

It follows that the (/>n are eigenfunctions simultaneously of A and :F. Using formula (2.218), we can then easily show that

(2.221) :F ((/>n) = (/>n = in(/>n (i2 = _1),51 n EN.

Remark 23. In quantum physics,52 the hamiltonian of the harmonic oscillator is defined by:

(2.222) li2 d2 1

Yf = - - - + - mw2q2 2m dq2 2

where m = mass of the particle, Ii = Planck's constant/2n, w = angular frequency of the oscillator. The preceding account shows that the eigenvalues En of Yf (called "energy levels" of the particle) are given by:

(2.223) En = (n + ~) liw, n EN.

The eigenvectors are the functions (/>n ( ~ q). o

50 See the Appendix "Distributions" in Vol. 2. 51 The unitary (non compact) operator :7 furnishes an example of an operator with a pure point spectrum consisting of a finite number of eigenvalues (i. -I, - i, I), each eigenvalue having infinite multiplicity. 52 See Chap. lA, §6.


7.5. Laguerre Operators and Functions

We assume here that Q = ]0, + CXl [ = IR+ and we put IR + = [0, Laguerre operator of order mEN defined on Q is

(2.224) d [ d ] [m - 1 x m2 ] 53

Lm = - dx x dx + -2- + 4" + 4x I,

where I is the identity. We shall first examine the case m = O.

69

+ CXl [. The

7.5.1. Spectrum of the Laguerre Operator of Order Zero. Laguerre Polynomials. We have

(2.225) Lo = - ~ [x ~ ] + (~ - ~) I . dx dx 4 2

Put

(2.226)

Then for u, v E .@(IR+), we note that

(2.227) def r [ du dv (x 1) ] ao(u,v) = <dou,v) = JIhl+ x dx 'dx + 4" +"2 uv dx.

We shall also use here the following notation:

(2.228)

Equipped with the natural norm, this is a Hilbert space. Note from (2.147) that

(2.229)

so that we can give V the norm defined by:

(2.230)

which is equivalent to the natural norm. We now have

Proposition 27. i) .@(IR+) is dense in V (defined by (2.228)), ii) the injection of V into H = L 2(1R + ) is compact.

Proof Part i) This will be conducted in four steps by demonstrating a succession of lemmas. I st step. Truncation at infinity. We prove

Lemma 4. The functions in V which vanish in a neighbourhood of + 00 are dense in V.

53 This operator turns up in the calculation of energy levels and eigenfunctions of the hydrogen atom (see Messiah [1]).


Proof of Lemma 4. In effect, if 0 is the function continuous on IR + which satisfies

(2.231) { O(x) = 1, for 0:::; x :::; 1

o affine for 1:::; x :::; 2

O(x) = 0, for x ~ 2 ,

then we consider OR defined by

(2.232) R > 0, 0R(X) = o( i) for all x E IR +;

then UR defined for given u, U E V, by:

(2.233)

satisfies

(2.234) {i) UR E V, V R > 0

ii) UR -+ U in V as R -+ 00 .

Verification of (2.234). i) This part is immediate, bearing in mind that u~ = OR U' + O~ u;

ii) r (1 + x)luR - ul2dx:::; 2f (1 + x)lul2dx -+ 0 as R -+ +00; J~+ J1XI ;;. R

finally, using the equality u~ = 0RU' + O~u

f xlu~ - u'I 2 dx :::; 4 r xlu'(xWdx + 2~ r xlu(xWdx ~+ J1XI ;;. R R J1XI;;' R

and the second term of this last inequality tends to 0 as R -+ 00 (here M = sup IO'(x)1 = 1), whence (2.234)). 0

X E ~+

2nd step. We now prepare the way for a truncation in the neighbourhood of zero by first of all showing that we can restrict ourselves to bounded functions.

Lemma 5. The functions in V which are bounded and vanish in a neighbourhood of + 00 are dense in V.

Proof of Lemma 5. Let u E V vanish for x sufficiently large. For kEN * introduce functions Uk defined by:

(2.235) {kif u(x) ~ k

uk(x) =. u(x) if lu(x)l:::; k· a.e. x

- k if u(x):::; - k

and let r/lk be the characteristic function of

(2.236) Ek = {x E IR+; lu(x)l:::; k a.e. x}.


We straight away prove that the derivative in the distribution sense of Uk is given by:

(2.237) u~(x) = I/Ik(X)U'(x) a.e. x E IR + .

Then

(2.238) { for all kEN *,

Uk E V, Uk is bounded and vanishes for x sufficiently large.

Using Lebesgue's thorem, we then have:

(2.239)

~Uk --> ~u in L2(1R +)

t+ 00 xlu~ - u'1 2 dx = t+ 00 x(l - I/Ik)2Iu'(xWdx --> 0

as k --> 00

whence the lemma.

3rd step. Truncation in a neighbourhood of zero.

o

Lemma 6. The functions in V with compact support contained in IR + are dense in the set of functions in V which are bounded and vanish for x sufficiently ·large.

Proof of Lemma 6. Let p be the continuous function defined by:

(2.240) {

p(x) = 0, for 0 ~ x ~ 1

p affine for 1 ~ x ~ 2

p(x) = 1, for x ~ 2 .54

For n E N*, we define Pn by:

(2.241) Pn(x) = p(nx) ,

so that

(2.242) p~(x) = np'(nx)

. [1 2J . [1 2J vamshes for x rt ;;';; and IS equal to n for x E ;;';; .

Now let v E V be bounded and vanish for x sufficiently large. The function Vn

defined by

(2.243)

is in V and has compact support contained in [~, + 00 [ .

54 Thus p = I - () with () defined by (2.231).


It is immediate (see 1st step) that

~v. -+ ~v 10 L2(1R+) as n -+ 00.

In an analogous manner, since v~ = Pnv' + P~v, we have

JxPnv' -+ Jxv' in L2(1R+).

It only remains to examine the behaviour as n -+ + 00 of the term Jx p~v. Now:

2 f: n2 xlv(xW dx, n

and v being bounded, we have:

IJxp~vl~2(~+) ::;; ~ sup Iv(xW; x E jR:+

hence Wn = Jx P~v remains in a bounded set in L 2(1R+); we can thus extract from {w.} a subsequence {wv } which converges weakly to WE L2(1R+). But Wn -+ 0 in the sense of ~'(IR +) as n -+ + 00 hence W .= 0. 55

Thus Vn -+ v in V weakly and we have already established density with respect to the weak topology. This in turn implies the density we seek to establish. Because if there exists Vo, orthogonal to the space of the u E V with compact support with respect to the scalar product in V, then ((vo, v~» = 0 for all n where v~ is the preceding approximation to Vo by truncation and regularisation of Vo.

Since v~ ~ VO weakly in V,

whence Vo = O. o

4th step. Regu/arisation. From steps 1 to 3 we deduce that the functions in V with compact support in IR+ are dense in V. Thus it suffices to approximate such a function v by a function of class ri OO • Hence let <f1" E > 0, be such a function E ~(IR) such that

L <f1, dx = 1, supp <f1, E [ - 1, + 1] .

The reader may verify without difficulty that <f1, * v -+ v in V (where v E V has compact support in IR + and * is the convolution product) as E -+ O. 0

55 In effect, we have:

n fin j~v(x)<p(x)dx = 0 for <pE!0(IR+), nsufficientlylarge. lin


Part ii). Note that we have:

(2.244) { Cfo: ~ > 0, u Ef~ 1 f + <Xl

lul 2 dx ~ lul 2 dx + -- (1 + x)lu(xWdx. o 0 1+R R

Now, for all given B > 0, we can find R so large that II u II

(2.245) 1 f+<Xl -1-- (1 + x)lu(xWdx < B.

+ R R

~ c implies that

Hence it all comes down to showing the compactness of the injection of

VR = {Jxu E L2(0, R), Jxu' E L2(0, R)}

into L2(0, R).

But if, for u E VR , we have

(2.246) {for all1} > 0, J: lu(tWdt ~ 8(1})lIull~ 8(1}) -+ ° with 1},

then it all comes down to showing that

B = {u E VR ; Ilull ~ C, suppu C [1}, R]}

is a compact set in U(1}, R). Now B is a bounded set in Hl(]1}, RD and hence is relatively compact III

L 2(]1}, RD, whence the result will follow once we have verified (2.246).

Verification of (2.246). If suffices to work with functions of class ct'l on [0, R]. If u is such a function then we have:

u(x) = u(r) + r u'(O') dO' ,

whence

lu(xW ~ {lu(rW + ILOg~1 f:' 100u'(O'WdO' J. Integrating first with respect to r E [0, R], then with respect to x on (0,1}), we obtain:

(2.247) { J: lu(xW dx ~ ~ luli2(O.R) + v(1})IJxu'li2(O.R)

v(tf) -+ ° with 1},

whence the result (2.246). o


Remark 24. From Remark 18, it follows that for u E V, we cannot in general define the trace at the origin, u(O). 0

Remark 25. E&(IR+) being dense in V, the dual V' of Vis a space of distributions. If FE V' then, from Riesz's theorem, there exist two functionsh,;; E L2(1R+) such that

(2.248) ( F, v) = t+ 00 (h Jx :: +;; Jx+l v ) dx for all v E V

(,) the duality between V' and V), the representation (2.248) being non-unique. On putting

gl = Jxh, g2 = J1+X;; , and on putting v = (() E E&(IR +) in (2.248), we then deduce

(2.249)

(2.249) characterises the elements of V'. o The bilinear form ao(u, v) defined by (2.227) is continuous and symmetric on V x V and coercive on V. From the Lax-Milgram theorem (see Chap. VI, §3 and Chap. VII, §1):

{ there exists a unique solution u E V of (2.250)

ao(u, v) = (f, v) (f given in V') for all v E V.

IffisgiveninH = L2(1R+),then(f,v) = ( fvdxanduED(do)(slogiven J~+

by (2.226». The following proposition characterises D ( do).

Proposition 28. The operator do defined in (2.226) is an isomorphism of

(2.251) D(do) = {u E Hl(IR+); (xu) E H2(1R+)}

onto U (IR + ).

Proof If u E V is such that v ---+ ao(u, v) is continuous on V for the topology of L 2 (IR +) then do u E L 2 ( IR + ). Since u is already in L 2 (JR+) from (2.229), we have

(2.252) - - x - + - E L (IR ). d (dU) xu 2 + dx dx 4

Let us, for the moment, assume

Lemma 7. If u satisfies (2.250) with fEU (IR +), then

(2.253) Jx u E V . o


From (2.252) and (2.254) we deduce

(2.255) d (dU) 2 + dx x dx E L (IR ),

which, by applying Lemma 3 (Hardy's), implies

du dx E L2(1R+);

hence

(2.256)

Furthermore:

d du dx (xu) = x dx + U E L2(1R +) from (2.254) ii) ,

d2 d (dU) du dx 2 (xu) = dx x dx + dx E L 2(1R +) from (2.255) and (2.256) ;

75

hence xu E H2(1R+) which proves the proposition, once we have proved Lemma 7.

Proof of Lemma 7. Introduce the function OR defined by (2.232) and put

(2.257)

We have

(2.258) { UR E L2(1R+) , uR -+ Jxu in U(IR+) as R -+ 00

VR E V.

We may thus take v = VR in (2.250) (withfE L2(1R+)). An immediate calculation shows that .


(2.259)


As JX :X (JX OR) = 0; + xO~ is bounded on IR + independently of R,

f + 00 (d )2 f + 00 o u2 JX dx (JXOR) dx ~ C 0 u2 dx (C = constant).

Moreover, Ve > 0, there exists a constant C(e) such that

I t+ 00 0Rf. JX uR dxl ~ C(e) t+ 00 f2 dx + e t+ 00 X(UR)2 dx

so that in virtue of the coercivity of the form ao(u, v) on V, we obtain

(2.260) II uR II ~ constant (independent of R)

for e sufficiently small. Thus we can extract a subsequence {UR'} from the sequence {u R } which converges weakly to w in V.

From (2.258), w = JX u, whence Lemma 7. 0

Remark 26. As in the case of the Legendre operator, we can again show that do is an isomorphism of

(2.261)

onto Hk(IR+), kEN. It follows that if the givenfis in tt' 00 (IR + ), then so also is the corresponding solution u; do is thus an hypo-elliptic operator. 56

Spectrum of the operator Lo. The eigenfunctions of Lo are the same as those of do. From Remark 26, if <1>. denotes an eigenfunction of do (or of Lo), we have <1>. E tt'oo (IR +). We are thus led to study the infinitely differentiable solutions of the differential equation

(2.262) - (xy')' + (1- ~) y = ).y, A > 0 .

If we put

(2.263)

the equation satisfied by z is

(2.264) z" + (1 - x)z' + Az = o. 00

Studying a solution of (2.264) of the form z = L a.x· leads to the recurrence o

relation for the a.:

(2.265) n - A

a. + 1 = (n + 1)2 a. , n ~ o.

56 Note that here again (see Remark 19), with the regularity on the boundary, we have here a slightly stronger property.


If A of. p, pEN, (2.265) leads to a solution Zo expandable in an entire series throughout the complex plane and such that zo(O) = 1. If A = p, we obtain a polynomial of degree p. In every case, the Wronskian method leads to a general solution of the form:

(2.266) Z = Azo + BZl Logx

where Z 1 is an entire function with Z 1 (0) of. 1. Such a solution is not acceptable if B of. 0 because Z 1 Log x does not belong to rcOO([R+ ). Thus, necessarily, B = O. Furthermore, if Zo is composed of infinitely many terms, the solution Yo satisfying (2.263) is not in V. The only admissible solutions of (2.264) are given by

(2.267) An = n, n E N

and are polynomials of degree n, the coefficients of which are given by (2.265). Another way of obtaining these polynomials is to use the method employed earlier for the Legendre and Hermite polynomials. The study of an analytic function tf; such that

zn(x) = -21 . r ex~t/J(¢)d¢ nl Jc)'

shall be a solution of (2.264) (with A = n) leads to us taking

(2.268) .1.(1') (¢ - 1)n C . I d h .. '1''' = ¢n + l' )' a clrc e centre at t e ongm;

we thus have

_ 1 r x~(¢-I)" . zn(x) - 2ni Jc)' e ¢n+ 1 d¢,

on putting 1 - ¢ = a, we obtain

(2.269) {

1 r ex(l - 11)

zn(x) = 2ni Jc)' an (a _ l)n + 1 da

C')' a circle surrounding the point (J = + 1 described in

the direct sense .

We then deduce

(2.270) 1 dn

Z (x) = - eX - (xne- X) . n n! dxn '

Zn is the Laguerre polynomial of order n. The function cP n defined by

(2.271)

is the associated Laguerre function.


We verify that

(2.272) t+ 00 cP. (x) cPm(x) dx = 15::' (15::' the Kronecker symbol) .

The Laguerre functions form an orthonormal basis for L 2 (IR + ), and a total orthogonal system in V. If we put

(2.273) d·

P ( ) _ , () _ x _ ( • -X) • X - n.z. x - e d x e , xn

p. satisfies the relation:

(2.274) p. + 1 - (2n + 1 - x)P. + n2 p. _ 1 = 0, n ~ 1 ,

and we can again show that we have:

(2.275) 1 a 00 x·

-- e- r-cx = L - Pn(x) , for Ixl < 1, 1 - x 0 n!

a e- r=x-Thus G(x, t) = --- is the generating function for the Laguerre polynomials. In

1 - x effect, we have, for Ixl < 1

00 (_ l)k tk Xk

G(x, t) = k~O k! (1 _ xt+ 1

00 (_ It 00 (n) L -- tk L xn k = 0 k! • = k k

f t (_It(n)xn~k, • = 0 k = 0 k k.

so that 00 x.

G(x, t) = L K.(t),- for Ixl < 1 , n = 0 n.

with

K.(t) = t (_I)k (n) n! tk = eX dRn (tne- I )

k = 0 k k! dt

as is shown by the Leibniz derivative formula. o Remark 27. We can again note (as elsewhere in the case of the Legendre and Hermite polynomials) that the Laguerre polynomials can be obtained by starting with the sequence of polynomials 1, x, ... , xn, ... and using the Schmidt orthonormalisation process, in this instance in the space of functions square integrable with respect to the measure e-xdx denoted: L2(1R+, e-Xdx). 0

Remark' 28. If Y(x) is the Heaviside function, then the function

(2.276) Y.cP n (cPn defined by (2.271)) is in L2(1R).

By Fourier transformation we obtain:

(2.277) 1 f + 00 _ iXT ( - l)n (1 - iT )n

t/ln(T) = M:: Y(x)cP.(x)e dx = M:: (1 + . ). + 1 V 2n - 00 V 2n IT


we then deduce that the functions { l/J n} n ~ 0 form an orthonormal basis of the image

space in L 2(1R) given with the scalar product L f g dx 57 • 0

7.5.2. Laguerre Operators of Order m, mEN * . With the operator Lm defined by (2.224) we very naturally associate the symmetric bilinear form

(2.278) i du dv am(u, v) = x -d -d +

IR+ x X

Let us then denote by 1/ the space

(2.279)

equipped with the norm

(2.280)

1/ is a Hilbert space. We verify that

Ilull

-- + - + - uvdx. (m - 1 x m2) 2 4 4x

u (2.281) if u E 1/, then Jx E e(IR+)

In effect, from Lemma 3 (Hardy's)

and

f>U 12 dX ~ (f>JxuI2dxy/\f:IJxI2dX)1/2 forall a> 0

from Cauchy-Schwarz, whence (2.281). 0

Thus we find ourselves in the following situation:

(2.282)

( V the space defined by (2.228», so that the injection of 1/ C; L 2 (IR + ) is compact. Since the bilinear form am(u, v) is continuous on 1/ x 1/ and coercive on 1/ for all mEN *, we are within the field of application of §2.6. Thus there exists a discrete spectrum for the operator Lm. To determine the eigenfunctions of Lm, we may remark that if we successively differentiate m times the equation (2.264), we obtain for u = z(m) the equation

(2.283) xu" + (m + 1 - x)u' + (). - m)u = 0

57 If nEZ, the "'" form an orthonormal basis in U(IR).


which can be put into self-adjoint form

(2.284) - xm+ le- x - + xme-x(A. - m)u = O. d ( dU) dx dx

If we put

(2.285)

then v satisfies

(2.286) d ( dV) (1 - m x m2 )

dx x dx + -2- - 4" - 4x v + AV = 0 ,

which is an exact translation of

(2.287)

As a result we have

Proposition 29. i) The spectrum Am of the Laguerre operator Lm(m EN), defined by (2.224) and with domain (for mEN *)

{ d du (m - 1 x m2 ) 2 } D(Lm) = UEr; - dxxdx + -2- + 4" + 4x uEL (IR+)

is

(2.288) Am = {An = n, n ~ m} .

ii) The eigerifunction <P': associated with the eigenvalue An = n (n ~ m) is called the n-th Laguerre function of order m. It is given by

(2.289)

where

(2.290)

(Pn given by (2.273); cn,m constants such that I <P':I = 1).

7.6. Chebyshev Operator and Polynomials

The Chebyshev polynomials 1',. are polynomials of degree n which are constructed very simply beginning with the circular functions. Earlier we had noted (see §2.7.2,

d2 the Neumann problem for - -2 + Ion ]0, 1 [) that the functions:

dx

(2.291) <Pn(x) = ft cosnx, n EN

form an orthonormal basis in L 2 (0, n). Now <Pn(x) is a polynomial Tn(Y) in the variable y = cos x.


The orthonormalisation formula

(2.292) I tPn(x)tPm(x)dx = <>:

can then be written

f + 1 dy m

Tn(Y) Tm(Y) J 2 = <>n , - 1 1 - Y

(2.293)

and the {Tn} n E 1\1 form an orthonormal basis in the space of functions square

integrable with respect to the measure dy on ] - 1, + 1 [; Tn satisfies the Jl - y2 differential equation

(2.294) - ~[Jl dy

- y. - = n u, n EN, X E -1, + 1 , 2 dUJ 2 ] [ dy

and we can show, beginning with the definition, that:

(2.295) { T,,(y) = ~ (- It Jr-l-_-y~2 _dn (1 _ y2)n - t V -;; 1.3.5 ... (2n - 1) dyn

nEN, YE]-I,+I[.

The functions Tn are naturally the eigenfunctions of an operator fJ, the Chebyshev operator, with compact resolvant, on the space

(2.296)

which is a Hilbert space for the natural norm. We will introduce the space

du (2.297) V = {u E L;(Q); (1 - y2)1/4 dy E L2(Q)} ,

which is a Hilbert space for the norm

(2.298) Ilull = (IUI~~ + L Jl - y21~:12 dy )I/2

The Chebyshev operator fJ will then be defined starting with the operator

(2.299) d = fJ + I L2 (I L2 identity in L;) p p

associated with the triple (V, H, t(u, v» ,

where

(2.300) t(U,v)~ In(Jl_y2dUdV + uv )dY, U,VEV, J" dy dy Jl _ y2

is a symmetric bilinear form continuous on V x V and coercive on V.


8. Application to the Spectrum of the Laplacian in U c 1R"

8.1. Dirichlet Problem for the Operator ( - .1)

Let Q be a bounded open set in [Rn. We take

(2.301) {V = H6(Q) , H = L2(Q)

n au av 58 a(u, v) = r (. L ::;-::;-) dx . J fJ 1 = 1 uXi uXi

The sesquilinear form a(u, v) is continuous, coercive on V and the injection of V into H is compact, so that, from the Lax-Milgram theorem and the results of Chap. VI, §3, we have

(2.302)

and

(2.303)

{ ( - .1) is an isomorphism of

D( -.1) = {u E HA(Q); .1u E U(Q)} onto U(Q)

{ G = (- .1)- 1 is continuous L2(Q) ~ D( -.1)

and compact L2(Q) ~ L2(Q) .

We can hence apply the theory from §7.2.6 to ( -.1) which is positive and selfadjoint:

(2.304) {the spectrum of ( - .1) in the Dirichlet problem

is thus a pure point spectrum.

The explicit determination of the eigenvalues and the eigenvectors of the Laplacian depend on the geometry of the domain Q. In general, it is out of reach. We now consider here some particular cases where the geometry of the domain is sufficiently simple to enable us to calculate explicitly the eigenvalues and eigenvectors by placing special emphasis on the methods (which depend on the invariance of the domain under a group of transformations, for example) which are more interesting than the result. More precisely, we propose - particularly for open domains Q-to determine explicitly the eigenvalues An and the eigenvectors Wn for the following problem:

find Wn E HA(Q) satisfying:

(2.305) i) a(wn,v) = An(wn, v) for all v E H6(Q)

ii) In 1 W n 12 dx = 1, (( u, v) = In uv dx ) ,

58 In this section we assume that the spaces are complex.


which is equivalent to the problem:

(2.306) -Llwn = AnWn' WnEH6(Q), LIWnI2 dX=1.

8.1.1. The Case of a Disc in jR2. Representations of the Group of Rotations in L2(jR2) and Separation of Variables. Bessel functions. We suppose here that Q is the unit disc in jR 2

(2.307) Q = {(x, y) E jR2; x 2 + y2 < I} ,

for which the boundary r = aQ is defined by:

(2.308) r = {(x, y) E jR2; x 2 + y2 = I};

Q is thus a regular open set in [R2 and the preceding theory applies. Thus we have to solve:

(2.309) { - Llw_" = AW" w ... lr - 0,

the fundamental property here being that the domain and the system (2.309) are invariant under rotations about the origin. This allows us to introduce the method of separation of variables. This method can be applied in a general fashion to all those physical problems for which there exists a compact group G of transformations leaving the problem invariant. The method of separation of variables rests on the study of the representations of the group G, here the group of rotations.

8.1.1a. Unitary Representations of a Commutative Compact Group. We commence by giving some definitions of a general character. i) Let G be a group not necessarily commutative; we denote by

(2.310)

the law of composition in the group. Let £ be a Hilbert space; 2 (£) the vector space of continuous endomorphisms59 of £. We denote by I 1£ (resp. ( , )£) the norm and scalar product in £.

Definition 4. A mapping

g f----+ o/Ig from G -+ 2(£)

such that

(2.311)

is called a representation of G in the Hilbert space £.

Definition 5. A representation g f----+ 0/1 9 is unitary if

(2.312) 1000gxJ.ff = Ixl£' for all x E £ .

59 That is, linear mappings of .ff into .K.


Definition 6. If G is a topological group, the representation 9 I---> ilIt 9 is said to be continuous if for all x E £, the mapping 9 I---> ilIt 9 x is continuous from G into £.

ii) Now let G be a commutative group in which we denote by + the law of composition.

Definition 7. A mapping

x: 9 ~ X(g) of G -+ IC

which satisfies for all g, g' E G:

(2.313)

(2.314)

x(g + g') = X(g)x(g')

\X(g)\ = 1 .

is called a character of G.

Note that from (2.313) it follows that

(2.315) x(O) = 1, X( - g) = X(g)

(i complex conjugate of z). If G is given the topological structure of a locally compact space for which the group operation is continuous, we denote by

(2.316) G the set of continuous characters of G.

It is easily verified that if we give G the law of composition obtained by taking the product of characters then

(2.317) G is a commutative group (the dual group of G) .

iii) Always assuming G to be a commutative group, let X be a character of G and £ an arbitrary Hilbert space; with the pair (X, £), we associate the representation

ilIt(X, g): ilIt(X, g)x = X(g).x for all x E £;

ilIt(X, g) E 2(£) for all 9 E G . (2.318)

then

(2.319) ilIt(x): 9 I---> ilIt(X, g) is a unitary representation of G .

Definition 8. ilIt (X, .) defined by (2.318), (2.319) is called a pure representation of type X.

The central theorem of the theory which interests us here is the H. Stone decomposition theorem (see e.g. Vilenkin [1]).

Theorem 8 (The Stone decomposition theorem). Let G be a compact commutative group; 9 I---> ilIt 9 a continuous representation of G in the Hilqert space £. Then i) £ is the direct Hilbert sum of the subspaces {£x; X E G}:

(2.320)


ii) each .Yex is stable under OUg :

(2.321)

iii) the restriction of OU g to .Yex is the pure representation of type x: (2.322)

85

S.l.lb. The Group T. Let T be the trigonometric circle (sometimes called the 'torus of dimension 1'), that is to say the unit circle given an orientation and an origin 0 for arcs; T is a compact topological space. We associate with ~ E T, the arc O~ which is a real number defined to within a multiple of2n. We identify the point ~ E T and the real number O~. The addition of arcs defines a commutative group structure on T. Given this structure, T is identified with the quotient group 1R/2nlf (If additive group of integers). Then it can be shown that:

(2.323) { the dual group ! of T is identified with If ; the dual group If of If is identified with T .

We shall now verify the Stone theorem for this example. i) Let .Ye be a Hilbert space and (h-+ OU II a unitary representat40n of T in .Ye. We can then define

(2.324) oUn E 'p(.Ye)

and we have

(2.325) { oU nO oU n' = 15:' oU n for all 15:' Kronecker symbol.

n,n' Elf.

(In effect we verify that

- - f2" f2" .. , , dOdO' OJtnoOUn, = OJtlloOUII,·e-InIl-1n1i --2-; o 0 4n

on putting 0" = 0 + 0', we see that

oUn ° oUn' = (21n J:" e- i(n - "')11' dO') (21n J:" OUII" e- i"/I" dO")

= 15:' oU n whence (3.325» .

ii) Denote .Yen = oUn(.Ye); then from (2.325) we deduce

(2.326) .Yen (\ .Yen' = {O} for n =F n' .

Now, for XoE.Ye, the function ({): OI-+OJt/IXO is -continuous with values in .Ye. Its Fourier coefficients are given by


we then easily verify that the series

L <pnrlnleinO, r E ]0, 1 [ nElL

converges uniformly on [0, 21£] to the function ({J in the space C6'°([0, 21£]; J'l') of

continuous functions with values in J'l' given the norm sup I ({J(O)I)f" o E [0, 27[]

In particular, for 0 = 0,

(2.327)

Thus J'l' is the topological direct sum of the J'l'n. iii) To calculate the restriction of Olio to J'l'n, let Xn E J'l'n, then from (2.325) Xn = oftnxn and

~ ( 1 f27[ ) '¥Ioxn = '¥Io'¥lnxn = '¥Io 21£ ° '¥I8'xne- inO' de' = ~ '¥Io + o,xne dO , 1 f27[ - in8' '.

21£ ,0

whence on putting 0 + 0' = 0"

/iiI einO f27[ /iiI - inO" da" = einO/ii", x inO -UOXn = h ° -uo"xne u - -Un n = e Xn ,

which establishes not only (2.321), the stability of J'l'n under '¥Io, but also (2.322), o ~ Xn(O) = ein6 being a character of If. iv) It remains to show the orthogonality of the J'l'n for n # n'. This follows directly from the fact that the operators (oft n) are hermitian and because of (2.325) they are orthogonal projections onto orthogonal subs paces. A proof can also be given without making use of the oftn: in effect ('¥lox, '¥Iox'))f' = (x, x'))f' if x and x' E J'l', since '¥Io is unitary. If we choose x E J'l'n' x' E J'l'n" we get: (einOx, ein'Ox'))f' = (x) x'))f', whence (x, X'))f' [1 - ei(n -n')o] = 0 for all 0 E If, so that (x, x'))f' = O. Thus we have Theorem 8 in the case where G = If. 0

8.1.1c. Representation of the Group of Rotations on L2(1I;t2). We now allow the group If to operate on 1R2. For this we introduce the rotation matrices:

;JA0 = (c~s 0 - sin 0) . (2.238) SIO 0 cos 0

Let J'l' = L 2 (1R2); we define a representation

o ~ '¥Io of If on L2(1R2) by:

(2.239)

60 Note that we could also have used the representation oii. defined by:

(CfI.f)(x) = f(M; I x).


As rll8 preserves the measure dx = dX 1 dX 2 on [R2, it follows that

(2.330)

Hence (0Zt8 ) defined by (2.329) is a unitary representation of 1f in L2([R2). From Theorem 8, (0Zt8 ) is the direct sum of pure representations. We then denote by L;([R2) the subspace of L2([R2) corresponding to the pure representation defined by the character Xn: e f--+ ein8, n E Z. We have

(2.331)

In polar coordinates, (2.331) is equivalent to

(2.332)

and Theorem 8 gives

(2.333) nEE

Now let Q be the unit disc defined by (2.307). The space L 2(Q) = U(Q; dx) is identified with a subspace of L 2 ([R2). This subspace is stable under the action of 0Zt 8'

Once again denoting by 0Zt8 the restriction of OZ/.8 to L2(Q) and on putting

(2.334)

we see that the representation 0Zt 8 decomposes into pure representations corresponding to the L;(Q).

8. 1.1 d. The Method of Separation of Variables. The system (2.309) being invariant under rotation, each proper subspace v., of ( - ,1) in the Dirichlet problem (2.309) is invariant for the representation 0Zt8 •

Again denoting by 0Zt8 the restriction of0Zt8 to V)., Theorem 8 permits the decomposition of the representation 0Zt 8 of 1f into pure representations. We then deduce

(2.335) { there exists an orthonormal basis of L2(Q) simultaneously diagonalising ( - ,1) and 0Zt 8 .

Let w)..n E L;(Q) be an eigenfunction, then

(2.336) { - ,1w).,n = ),w)..n

w)..n E H6(Q) n L;(Q) .

From (2.332), we have in polar coordinates

(2.337)

Since the Laplacian can be written in polar coordinates as

02 1 0 1 02

,1=-+--+--Op2 P op p2 oe2 '

(2.338)


so (2.336), taking into account (2.338), gives

(2.339)

')" 1, ( , n2) 0 1 V;. n + - VA n + I\. - -2 V;. n =

. p' p'

ii) V;',n E Hl(]O, 1[)

iii) v;.,n(1) = 0 .

o Remark 29. The method of separation of variables consists of replacing a partial differential equation (here equation (2.309)), on utilising the symmetry properties (here the invariance under rotation), by a partial differential equation involving fewer variables (here the ordinary differential equation (2.339) i)). We have thus justified here the classical method which, in order to find the solutions of (2.309), consists in putting

w;.(p, e) = I/t ;.(p)cp;.(e) ,

and noting that the equation

I/t~ + ~ I/t~ p 1 cp~(e)

+ A. = - p2 cp;.(e) ,

separates into two second order equations, one in the variable p the other on the variable e. 0

Remark 30. We have made use only of the invariance of Ll and Q under rotation. The same method can be applied to any elliptic operator A of order 2 which is invariant under rotation; we would then get the analogue of system (2.339), the differential equation (2.339) i) then being replaced by another 2nd order differential ~~~ 0

Remark 31. Let us further note that the Laplacian being an operator with constant coefficients possesses another remarkable property: that of being invariant under translations of /R 2 .

In wishing to take account of these two properties: invariance under rotation and invariance under translation, we are led into making an analogous study of the group of displacements of /R 2 denoted by M(2). This group is noncommutative. For this point of view, one may refer to the previously cited work by Vilenkin. 0

Remark 32. Let us consider the decomposition of L2(/R2) asserted by Theorem 8:

L2(/R2) = EB L;(/R 2). n E 7L

(2.340)


We know that ff' is an isomorphism of U(jR2) onto L 2(jRZ). It is easy to check that

(2.341)

Thus, iff is written in polar coordinates

f(p,O) = u(p)ein8 , fE L;(jR2) ,

we have: l(r, 0') = v(r)ein9' ;61

then v (Hankel transform 62 of order n of u to within a coefficient ( - it) is given by:

(2.342) f+OO

v(r) = (- it 0 u(p)Jn(pr)pdp

where J n is defined by

(2.343)

Proof of (2.34 J). If ~9 is the rotation matrix defined by (2.238), we first note that:

~9X.y = X'~-9Y (scalar product in jR2),

so that if we put

and observe that the measure dX 1 dx z is invariant under rotation, we immediately have, beginning with (2.340),

(2.344)

ButfE L;(jR2) is equivalent to

fo = ein8 f,

which from (2.344) is equivalent to

(1)9 = ein9 j whence (2.341).

Proof of (2.342). We write (2.340) in polar coordinates:

v(r) = ~ f27t f+ 00 u(p)einge-iprcos9 pdp dO. 2n 0 0

Using Fubini's theorem, we may first integrate with respect to 0; we get:

f + 00 [ 1 f21t.. ] v(r) = 0 u(p) 2n 0 e,nge-,prcos9dO pdp

61 Where (r, 8') denotes the polar coordinates of the dual variable y of x. 62 See Chap. IlIA, §3.

o


whence (2.342), In(t) being defined by (2.343) following the change of variable

e = 3n - e' 0 2 .

Definition 9.63 The function J n defined by (2.343), namely

In(t) = ~ f2" e-in8e+itsin8 de 2n 0

is called the Bessel function of order n.

8.1.1e. Properties of Bessel Functions. Let us first of all note that (2.343) is none other than the coefficient of order n in the expansion in e(O, 2n) of:

(2.345) nEI

The function e I-> eitsin8 is continuous in e and of class ~oo for fixed t and hence belongs to L2(O, 2n) (thus one knows that {e in8 }, nEZ, forms an orthonormal basis for L2(O, 2n) for the scalar product

1 f21t (f,g) = 2n 0 fgde).

Further, because of the regularity of this function, the convergence of the series on the right-hand-side of (2.345) is uniform in e for fixed t. The function e I-> e it sin 8 is called the generating function for the Bessel functions. It is possible to develop a trigonometry for Bessel functions; we have:

(2.346)

(2.346) is an 'addition formula'.

Proof of (2.346). We have

rEI

ei(t+t')sin8 = eitsin8eit'sin8 = ( L eiP8 J p(t))( L eiq8 Jq(t'))' pEl qEI

These two series being absolutely convergent, we may form their product term by term and then re-arrange it to give:

e i (t+t')sin8 = L ei8 (P+q)J p(t)Jq(t') = L eir8 L Jp(t)Jq(t') ~qEI rEI p+q=r

rEI

Then multiplying the two members by e - in8, and integrating term by term with respect to e, we get (2.346). 0

63 See Chap. IlIA, §3.2.


We further have:

(2.347) {

i) J-n(t) = (-ltJn(t)

+ 00 (_ 1)P ( t )n + 2p ii) In(t) = L -, n ~ 0 .64

p=o(n + p)!p! 2

We also note that I n has the same parity as n, and that

(2.348) d _ 1 .64 dt I n - 2: (In-l - I n + I ),

it can then be shown that J n admits an infinite sequence of zeroes {J1.n. q } q E 7L for which an estimate is given by the formula:

(2.349) lJ1.n,Q - n(~ + ~ + q)1 < ~I where Cn is a constant, depending on n and where q E 7L The formula (2.349) follows from the asymptotic expansion corresponding to It I ~ + 00 (due to Poisson):

(2.350) ( 2 )1/2 ( 1 n) e(t) J (t) = - cos t - -nn - - +-n nt 2 4 t l/2

where e(t) ~ 0 as It I ~ + 00, n being fixed, Hence,for sufficiently large Itl, the zeroes of J n are close to those of

cos (t - n2n - ~)

(see also §2,9.2 below, application of the Min-Max formula). Finally, it is easy to show that I n satisfies the second order differential equation

(2.351 )

called the Bessel equation of order n.65

8.1.1£. Spectrum of the Laplacian in the Dirichlet Problem for the Unit Disc and Bessel Functions. From (2.351), a solution of (2.339) i) is provided to within a multiplicative constant by

(2.352)

Putting v = uJn in (2.339), we obtain a first order differential equation in u' which is integrable by quadrature. On studying the solutions so obtained, we notice that the only solutions which are

continuous at p = 0 are of the form CJn(viip).

64 See Chap. IlIA, §3. 65 See Chap. IlIA, §3.


The boundary condition (2.339) ii) then shows that we must have:

(2.353)

The spectrum of ( - ,1) in the Dirichlet problem with Q defined by (2.307) (unit disc) is then

(2.354)

where {Iln,q}q E N are the zeroes of In. To each eigenvalue An,q = Il;,q there corresponds a proper eigenspace generated by Wn,q and w_ n,q where

(2.355)

Cn,q being determined by In I wnjp, ew pdp de = 1.

The multiplicity of each eigenvalue An,q (see § 1.1, Remark 2) is thus equal to 2, except for the first eigenvalue (n = 0) which is simple.67

8.1.2. The Case of a Cylinder in 1ij3. We now consider the problem (2.309) in the case where Q is the cylinder in 1ij3 defined by:

(2.356) Q = {(x, y, z) E 1ij3; x 2 + y2 < R, 0 < Z < I} .

In this section and those that follow, we will develop simple considerations analogous to those in the preceding case without going any deeper into the underlying groups. Let

(2.357)

Denote by

{ ,1x,y,Z be the Laplacian in 1ij3 corresponding to the

system (2.309) for Q given by (2.356) .

(2.358) u { -,1z = - :

Z

22 the Laplacian in z in the Dirichlet

problem for ]0, l[ .

The problem (2.309) corresponding to Q given by (2.356) is invariant for ( - ,1z) (i.e. ( - ,1z) commutes68 with ( - ,1x,y,z)); we then have the analogue to Theorem 8: the space L2(Q) can be decomposed into a direct hilbert sum of the form

+00 (2.359) L2(Q) = EB L;(Q)

1

where

(2.360) L;(Q) = {f E L 2(Q); f(x, y, z) = rp(x, y). wn(z)}

66 In effect fl.': n" = /1; .• which (for the eigenvalues) means we need only consider n ;;, O. 67 This is in accord with the Krein-Rutman theorem (see the appendix to this Chap. VIII). 68 See later on.


and where Wn is the eigenfunction of - Az for the Dirichlet problem for ]0,1 [, given by:

(2.361) ( ) _ fi . nnz N* Wn Z - '1/ I sm -1-' n E •

We are thus led to study the eigenfunctions of the form:

(2.362)

which are common to (- A z ) and to (- Ax,y,z)'

Then ({J;., n satisfies:

(2.363) { ( n2n2)

- Ax,y({J;.,n = A. - T ({J;.,n

({J;',n E H6(D R )

where DR here denotes the disc with centre 0 and radius R. We thus find ourselves coming back to determining the ({J;.,n as in Sect. 8.1, on putting

n2 n2 112 = A. - --

/2 (2.364)

The spectrum is thus

(2.365) n > 0, p ~ 0, q EN} where II p , q is a zero of the Bessel function J p and the eigenfunctions are of the form

(2.366) .1. (e) - C J ( ) ;po . nn 69 'l'n,p,q p, ,Z - n,p,q p IIp,qP e sm T Z ,

Cn,p,q being chosen in such a way that

(2.367) L lI/In,p,ql2dxdydz = 1 .

8.1.3. The Case of a Parallelepiped in 1R3. Here we are interested in the problem (2.309) in the open set Q c 1R3 defined by

(2.368) Q = {O < x < a, 0 < y < b, 0 < Z < c} ;

we note that (with analogous notations to those in (2.358»

(2.369)

It follows from considerations analogous to those made in Sect. 8.1 that (- A), (- Ax), (- Ay), (- A z ) commute 70 and admit a common system of eigenvectors.

69 We must not forget that for the eigenfunctions o/I •• P •• , we should take P E Z (and not only p ;" 0). 70 See further on.


The method of separation of variables, analogous to that developed in Sect. 8.1, then leads to the spectrum:

(2.370) { Pn2 12n2 m2n 2 } A = 7 + V + ~; k, I, m E N*

and to eigenfunctions:

(2.371) ( 8 )1/2 . knx . Iny . mnz Wk I m = -b sm - sm -b sm--.. a cae

which form an orthonormal basis in £1(Q) and H MQ), Q defined by (2.368).

8.1.4. The Case of a Ball in 1R3. Spherical Harmonics. We are now dealing with the open set Q in 1R3 defined by

(2.372) Q = {(x, y, z); x2 + y2 + Z2 < I} .

We observe that (- A) and Q are then invariant under the group of rotations about the origin of 1R3 (which is a non-commutative group).71 The representation theory in L 2 (1R3) of thii group ofrotations which preserve the unit sphere (and for which we refer to Vilenkin [1]) permits the development of considerations analogous to those in Sect. 8.1 and justifies the method of separation of variables in the Dirichlet problem for the Laplacian in Q defined by (2.372). Denoting by S2 the sphere in 1R3, we first observe that in spherical coordinates (r, 0, cp) we have

(2.373) {L2(1R3) = L2(1R+, r2dr) ® L2(S2, de)

where e = (0, cp),de = sin OdOdcp .

Then the representation theory for the group SO(3) shows that L2(S2, de) decom-poses into

+00 (2.374) L2(s2, de) = Ee ~

,~O

where ~ is a vector subspace of finite dimension

(2.375) dim ~ = 21 + 1.

More precisely (see Vilenkin [1] and also Miiller [1]), ~ is the eigenspace corresponding to the eigenvalue - l(l + 1) of the Laplace-Beltrami operator or the Laplacian on the sphere given by:

(2.376) 1 [0 0 0 (1 0)] 72 A B = sin 0 00 (sin 0) 00 + ocp sin 0 ocp .

71 CaIled 80(3), 8 for symmetry, 0 for orthogonal, 3 for [R3.

72 In quantum physics (see Chap. lA, §6, and Messiah [I], vol. I), we sometimes use the notation

o L z = -i-. ocp


Then

(2.377) { V; = the mv~cto~sp~ce ~enerated by . {Y/ ,m - I, I + 1, ... , +/}, lEN,

95

the functions Y;": e -> Y;"(e) are called spherical harmonics; they are the eigen

functions common to the operators - LIB and - i ~ and we have aq>

(2.378) {-LIB Y;" = l(l + 1) Y;"

ia - aq> Y;" = mY;" ;

{ Y;"}',m forms an orthonormal basis for U(S2, de). For m ~ 0, we have:

(2.379)

where the P;" are the Legendre functions defined in Sect. 7. Hence the space L 2([R3) decomposes into:

(2.380)

The method of separation of variables for (for example) the study of the spectrum of - LI in the Dirichlet problem in the euclidean ball leads then to:

[ a2 2 a 1 J -Llf(r)g(e) = - ar2 + ~ ar + r2 LIB f(r)g(e) = Af(r)g(e) ,

and hence leads to a solution of the problem:

{ + a2f + ~ af + [A _ ~ + I)Jf = 0

ar2 r ar r2

f(l) = 0 .

(2.381)

By a change of variable p = j'E, we are led to solve the differential equation:

[ d2 + ~ ~ + 1 _ l(l + I)J Y = 0 dp2 P dp p2

for which the solutions regular at the origin are called spherical Besselfunctions and


denoted by j/. We have:

(2.382) j/(p) = C: y/2 J/+ 1/2(P)

where J/+ 1/2 is the Bessel junction of order I + 1/2. The eigenvalues A of the problem which interest us are thus the roots of the equation:

(2.383)

and the solutions of (2.381) are given by

f,,(r) = j/( JI:, r), An given by (2.383) .

o

8.2. The Neumann Problem Relative to the Laplacian

Let Q be a connected bounded open set in [Rn with a sufficiently regular boundary. We take

(2.384)

v = HI(Q), H = L2(Q)

a(u, v) = ao(u, v) + (u, v)

(u, v) = In uv dx .

The sesquilinear form ao(u, v) is continuous on V x V, but is not coercive on V. On the contrary, the sesquilinear form a(u, v) satisfies the hypotheses of the Lax-Milgram theorem. Hence we define an operator A in L2(Q) with domain

D(A) = {u E HI (Q); v ~ a(u, v) is continuous on HI (Q)

for the topology of L 2 (Q) }

oul = o} on r

and such that

a(u, v) = (Au, v) Vu E D(A), Vv E Hl(Q).

Recall that the operator A associated with the triple (H 1 (Q), L 2( Q), a) is an isomorphism of D(A) onto L 2(Q) (see Chap. VII, §1). Since (if Q is regular enough) the injection of HI(Q) into L2(Q) is compact, we have:

(2.385) G = (-.1 + I)-I is compact and self-adjoint from U(Q) ~ U(Q),

(2.386) -.1 + I is an isomorphism of D(-.1) onto L2(Qy,


where

(2.387) D( -,1) = {uEHl(Q); v H a(u, v) is continuous on Hl(Q)

for the topology of L2(Q)} .

The spectral problem

{find ~;.EHl(Q)

(2.388) ao(u;., v) = A(u;., v) for all vEH1(Q),

which is equivalent to the Neumann problem

{find u;.EHl(Q)

(2.389) -,1u;. = AU;. ,

admits the eigenvalue )00 = 0 and we have:

(2.390) dim N ( - ,1) = 1, Vo = N ( - ,1) = kernel of - ,1

so that

(2.391) Vo is generated by the constant 1.

97

We have furthermore (see Chap. IV, Poincare's inequality) established the relation existing between the second eigenvalue Al and the Poincare constant. The explicit determination of the sequence {)o.} of eigenvalues is generally out of reach (see Chap. XII for the numerical calculations). In the case of simple geometries, the considerations of symmetries analogous to those made for the Dirichlet problem can be developed to justify a method of separation of variables. We restrict ourselves to an example.

The case of a rectangle in 1R2. We put

(2.392) Q = {(x, Y) E 1R2; 0 < x < a, 0 < y < b}

and we denote:

(2.393) a2

(- ,1x) = the operator - ax 2 in the Neumann problem on ]0, a[ ,

with a definition analogous to (2.393) for (-,1 y), so that

(2.394) -,1 = (-,1x) + (-,1 y), and (-,1x), (-,1y) commute.

This allows us to illustrate the decomposition:

00

(2.395) L 2(Q) = EEl L;(Q) .=0

where

(2.396) L;(Q) = {fEU(Q); f(x, y) = cp(x) fi cos n~y} n ~ 0

98

where

(2.397)


fi nny . . . ;]2 y ~ V b cos b IS the n-th eigenfunctIOn of - oy2

in the Neumann problem in ]0, b[ corresponding to

n2n 2

the eigenvalue [;2 .

On seeking within L;(Q) the eigenfunction w".n of A with

we are led to determining the qJ '" n as eigensolutions of the Neumann problem for 02

- ox2 on ]0, a[.

Hence we obtain the eigenfunctions wn , p and the eigenvalues An,p defined by:

(2.398)

There are, of course, analogous considerations for the problems of mixed type, etc. 0

9. Determining the Eigenvalues of a Self-Adjoint Operator with Compact Inverse. Min-Max and Courant-Fisher Formulas

In this §2.9 we put ourselves in the situation considered in §2.6 where an operator A is given by a continuous hermitian sesquilinear form a(u, v) in a Hilbert Space V and we seek to express the eigenvalues of A with the aid of the form a(u, v) by making use of ideas from the calculus of variations.

9.1. First Expression for the Eigenvalues

We take two Hilbert spaces Vand H with

{V c; H, the injection c; being compact

(2.399) V being dense in H ,

I I, ( , ) denoting respectively the norm and scalar product in H, and II II, (( , )) denoting respectively the norm and scalar product in V. Let a(u, v), u, v E V be a continuous hermitian sesquilinear form on V x V satisfying (2.73) and let A be a self-adjoint operator with domain D(A) dense in H defined by (2.74). Recall that A is an isomorphism of D(A), given the graph norm, onto H, with the compact (self-adjoint) inverse A -1 from H into H.


Our interest is in the spectrum of the operator (A, D(A)):

(2.400) Au = AU, U E D(A) .

Let

(2.401) E( ) = a(u, u) U luI 2

and

(2.402) S = {u E D(A) ; a(u, u) = I} .73

Theorem 9. i) There exists u* E S such that

(2.403) E(u*) ~ E(u) for all UES

ii) u* satisfies (2.400) and the eigenvalue A * associated with u* is given by:

(2.404) A* = E(u*).

Proof We first of all note that from continuity of the injection of V into H, there exists a constant c > 0 such that I u 12 ~ c II U 112 for all u E V so that from (2.73), we have:

( ) _ a(u, u) II ul1 2 rx 0 E u - ~ ~ rx W ~ ~ > , 'r/u E V.

Let us put

(2.405) ),* = inf E(u) UES

( we have A * ~ ~ > 0) . By definition of in/, there exists a sequence {ud kE 1\1, Uk E S, 'r/ k, such that

lim E(uk ) = A*. k ~ 00

Since A * i= 0, we have (taking account that a(ub Uk) = 1)

(2.406) A . *

From (2.405) we have:

with

a(uk + ZV, Uk + zv) _c:---,-__ --"--c,,--------'- ~ A* for all v E D(A) , Z E C ,74

IUk + zvl2

{a(uk + ZV, Uk + zv) = 1 + IzI 2a(v, v) + 2Re[Za(uk,v)] ,

IUk + zvl 2 = IUkl 2 + IzI21vl2 + 2Re[z(ub v)] ,

73 If we prefer, we can take a(u, u) for the norm of u in V, so that S is then the unit sphere. 74 In effect w = (Uk + zv)/(a(uk + zv, Uk + ZV))1/2 E S.


whence it follows that

(2.407) IzI2(a(v, v) - 2*lvI2) + 2Re[i{a(uk, v) - 2*(uk , v)}] + 1 - 2*lukl2 ~ 0

Taking Z E IR; we obtain a trinomial of second degree which is non-negative for all Z E IR so that we must have

Since lim 1- 2*lukl2 = 0, we deduce

lim Re{a(ub v) - 2*(ub v)} = 0 for all vED(A), k~ 00

and, on replacing v by iv, we also have

lim 1m {a(ub v) - 2*(ub v)} = 0 for all vED(A). k ~ 00

It follows that

(2.409) lim (AUk - 2*ub v) = 0 for all v E D(A) . k ~ 00

Since a(uk, Uk) = 1, we can extract a subsequence again denoted by Uk which converges weakly in V and strongly in H to an element denoted by u* and which satisfies (from 2.409)

(2.410) a(u*, v) = 2*(u*, v)

for all v E D(A), and by density for all v E V. The mapping v ~ a(u*, v) is thus continuous on V for the topology in Hand u* E D(A). Thus (2.410) implies that Au* = 2*u* and that u* is hence an eigenvector of A relative to 2* if u* i= O. Now Uk converges strongly in H to u*; hence

from (2.406), and Theorem 9 now follows.

Put u* = Ur. 2* = 21 and denote

(2.411)

o

By an argument analogous to that in the proof of Theorem 9, we deduce that there exists U2 E F2 which minimises E(u) on F2; U2 satisfies

(2.412)

where 22 = E(u2) = inf{E(u); u E F2} .

Since, on the one hand

(2.413)


and on the other

is such that ~ E F2 , we deduce from (2.412) and (2.413) that Ivl

(2.414)

101

so that )'2 is an eigenvalue of A associated with the eigenvector U 2 . We thus derive the following more general statement.

Corollary 2. Let {uJ, i = I, ... , n - 1 be the first n - 1 eigenvectors 7 5 of the operator A,76 and let:

(2.415) Fn = {u E S ; (u, ui) = 0, i = I, 2, ... ,n - I}, n? 2 .

There exist un E Fn which minimise E(u); un is an eigenvector of the operator A, and An' defined by

(2.416)

is the eigenvalue associated with Un.

Proof After an argument analogous to that above, there exist Un E Fn which minimise E(u) and which satisfy:

(2.417)

We show that (2.417) holds for all w in V. We determine the constants t), ... , t n -) in such a way that

(2.418)

Writing

gives:

We notice that:

(2.419)

V n-)

·h " -EFn WIt V = w + L.. tiui for WE V. I vi i=)

(v, UJ = 0, i = I, ... , n -

{ a(un, uJ_= 0, (un' UJ - 0,

i = I, 2, . . . ,n - 1

so that from (2.417) and (2.419), where we take v to satisfy (2.418):

a(un, W) = An(Un, w) for all w E V

which proves the corollary.

Remark 33. The eigenvalues obtained satisfy:

(2.420)

75 Obtained by the previous method. 76 With the hypotheses made at the start of this Sect. 9.1. 77 0 is not an eigenvalue.

o


in effect we have A1 inf a(u, 2U) ~ rx, because from (2.73) (if lIull ~ luI) UES lui

a(u, u) ~ Ctllul1 2 ~ rxlul 2 .

On the other hand, to find Un' we take the minimum of E(u) over the set contained in the set used to obtain the Un _ l' It can be shown that we then get all the eigenvalues and eigenvectors of the operator A. 0

The inconvenience of this method is that the determination of the n-th eigenvalue makes reference to all the previous n - 1 eigenvectors u l' u2 , .•. , Un _ l' We now proceed to derive an expression for the (n + l)-th eigenvalue with the aid of the form a(u, v) (or of the operator A), but without making any reference to the eigenvectors U b ... , Un'

9.2. The Min-Max Formula

We have seen that An = inf(E(u)), the inf being taken over the u E D(A) (or H), lui = 1, orthogonal in H to the preceding n - 1 eigenvectors u1 , ••. , un - 1 • Then let Vn - 1 be a subspace in V of dimension n - 1 and consider the quantity:

{Am(Vn - 1 ) = inf E(u), the irifbeing taken over the u (lui = 1).78

(2.421) orthogonal to Vn - b that is to say belonging to V ~-1 .

If Vn - 1 is generated by u1 , • •• , un - 1 , we have from Corollary 9 that

(2.422)

But as Vn - 1 varies amongst all the subspaces of dimension n - 1, the irif can be smaller. Thus we have

Theorem 10. The Min-Max formula. The n-th eigenvalue An of the operator A is given by the formula

An = max Am(Vn- d = m. ax {min a(u, u)} . Vn_lCV V~-lCV UEV~-1

I U 1= 1

(2.423)

Proof Let Wn be the subspace of H generated by the first n eigenvectors. Let Vn- 1

be a subspace of H of dimension n - 1; there then exists Uo E H such that

(2.424)

U' n

(In effect, we seek Uo = -I ? I with u~ = L Ci U i , and the condition u~ E V ~ - 1 Uo i= 1

imposes n - 1 linear conditions on the coefficients C l' C2 , .•. , Cn • Thus there is always a solution). We deduce

(2.425) inf a(u, u) ~ a(uo, uo) . UEV~-I lui = 1

78 In Am' m is the abbreviation for 'minimal'. Note that if U E H with U I/o V, we put E(u) = + 00.


n

Now a(uo, un) = (Auo, un) and Uo L CkUk; furthermore AUk = AkUk since Uk is k=l

an eigenvector of A. Hence

n n n (2.426) (Auo, un) = L L CkCjAk(Uk> u) = L AkCf·

k=l j=l k=l

Now we know that Al ~ A2 ~ ... ~ An and hence from (2.426)

n

(2.427) (Auo, uo) ~ An L cf = Anl uol 2 = An' k=l

From (2.425) and (2.427) we deduce

whence

(2.428)

now since Vn - 1

Corollary

inf a(u, u) ~ An' UE V ~ - I lui = 1

max inf a(u, u) ~ An ; Vn-lCV uevit-t

lui = 1

Wn- 1 = {U1, ... , un-d we have from Theorem 9 and its

inf a(u, u) = An , ueWia-l lui = 1

SO that the max is attained for Vn _ 1 = Wn _ 1, whence formula (2.423) and T~~mlQ 0

9.3. Corollary to the Min-Max Formula. Comparison of the Eigenvalues of Two Operators

The hypotheses on the spaces are the same as before.

Proposition 30. Let a(a, v) and b(u, v) be two hermitian sesquilinear forms continuous on V x V and coercive on V. We assume

(2.429) a(u, u) ~ b(u, u) for all u E V .

Denote by An(A) (resp. An(B)) the n-th eigenvalue of the operator A (resp. B) associated with a(u, v) (resp. b(u, v)), counted with their multiplicities. Then

(2.430) An(A) ~ An(B) for all n.

Proof This follows immediately from the formulae (2.423) and (2.429). 0

Corollary 3. Let W be a closed subspace of V dense in H. Denote by:

{Ad the sequence of eigenvalues of the operator defined by (V, H, a(u, v)) ;

{.ud the sequence of eigenvalues of the operator defined by (W, H, a(u, v)) ;


then we have 79

(2.431) Ak ~ fJ.k for all k.

The corollary being immediate, we give instead an example of this situation. Let Q be a regular bounded open set in [Rn. Put:

(2.432) {

H = U( Q), V = HI (Q), W = H 6( Q)

f ( n au au ) a(u, v) = .L ;-;- + uu dx.

Q ,= 1 uX j uX j

The operator associated with the triple (V, H, a(u, v)) is -A + I for the Neumann problem, that associated with the triple (W, H, a(u, v)) is -A + I for the Dirichlet problem. Therefore we can apply Corollary 3 to this situation. 0

We now turn to give an important application of the Min-Max theorem. Consider a variational situation (V, H, a(u, v)), the hypotheses being those in Sect. 9.1. Let b(u, v) be another hermitian form continuous on V x V; put:

(2.433) a1(u, v) = a(u, v) + b(u, v) ;

a1 (u, v) appears as the hermitian form a(u, v) 'perturbed' by b(u, v). We now make the hypothesis that this perturbation is 'small' in the following sense;

(2.434) { 3c > 0 such that

Ib(u, u)1 ~ c.lul 2 for all u E V.

Let (A, D(A)), (A l' D(A 1)) be the self-adjoint operators associated with the hermitian forms a(u, v) and a1 (u, v) respectively; they have a discrete spectrum;80 denote by Ak (resp. J1.d the eigenvalues of A (resp. A d arranged in order of increasing magnitude (with their multiplicities). Then we have

Theorem 11. If b(u, v) satisfies (2.434), then

(2.435)

with (see (2.434)):

(2.436) Irkl ~ c for all kEN.

Proof In effect, from (2.433) and (2.434) we have:

la1(u, u) - a(u, u)1 = Ib(u, u)1 ~ clul 2 •

Thus for u E V, lui 1, we have:

la1 (u, u) - a(u, u)1 ~ c .

79 We further assume that a(u, v) is a hermitian form continuous on V x V and coercive on V. 80 The reader should verify that for a 'perturbation' b satisfying (2.434), the operator A 1 still has a compact resolvant.


We deduce:

I max inf a1 (u, u) - max inf a(u, u) I :(; c , Yn -lcVueVA-l Vn-lcVUEV~-1

lul=l lul=l

whence (Max-Min formula)

and o

Applications of the Min-Max Formula

1. Sturm-Liouville Problem

Let us consider the non-degenerate Sturm-Liouville eigenvalue problem

(2.437) {

Un EdH(WO. :~~)' a > ° - dx k(x) dx + qUn = Anun

where k(x) > ° on [0, a], and q(x) > 0, k, q E !6'°([0, a]) are given. We seek an equivalent to An for fairly large n by employing Theorem 11. On putting

the problem (2.437) is transformed into:

(2.438)

where r is a given continuous function. On putting

(2.439)

from the Min-Max formula we get:

-IX dx t- 11.'

° y' k

(2.440) An = max mill a(v, v) .

Consider

(2.441)

Vn-lC¥ VEV~-l I vi = 1

r 1 dv 12 b(v, v) = Jo dt dt;


then, for Ivlz = 1, we have

la(v, v) - b(v, v)1 ~ r Irllvl2 dt ~ max Ir(t) I = R, Jo IE[O,I)

thus

I max min a(v, v) - max min b(v, v) I ~ R , Vn-lCV VEV~-l Vn-tCY VEV~-I

Ivl~l Ivl~l

or

(2.442)

where Jl.n is the n-th eigenvalue of

(2.443) d2wn - ili = Jl.nWn, Wn E Hb(]O, ID·

We know (see §2.7) that

(2.444) {2 . nn

Wn = \(i sm T t , n E N*,

so that (2.442) can be written

We deduce that

(2.445) n2 n 2

A.n = J2 + an, I an I ~ R .

Consequently, for sufficiently large n we have:

(2.446) n 2

A.n '" Cn 2 , C = constant = [2 .

Returning to the original Sturm-Liouville operator (given by (2.437)), we get

(2.447) n2 1 ( fa dX)2 lim .., = """"2 fi: = llC. n-ro II.n nov k

2. We have already calculated explicitly the eigenvalues of the operator

-,1 + I

for the Dirichlet and Neumann problem on simple geometric domains: rectangle, disc, ... Theorem 11 gives a first approximation to the eigenvalues of the operator

{ -,1 + qI, with q a real function,

(2.448) q E L ro (Q) .


In effect, for q E LOO(Q) and b(u, v) = L quvdx, we have (2.434).

3. The method further applies to degenerate problems. As an example of this application we shall use it to find the asymptotic behaviour of the zeroes of the Bessel function J q , q E N discussed in §2.8.1. We know that J q satisfies the differential equation:

(2.449) J~ + ~J~ + (1 - ::)Jq = 0, so that if we put:

(2.450)

W q satisfies

[ d2 4q2 -1] (2.451) - -2 + -"""--;;2- Wq = AWq . dx 4x

Let Q = JO, 1 [; introduce the SPilce:

(2.452) V= {UEH 1 (Q), ~EL2(Q), u(l) = o} ~ HMQ),

and put

(2.453) { H = L2(Q)

r (dU dv 4q 2 - 1 _) a(u, v) = JQ dx dx + 4x2 UV dx.

The (singular) operator associated with the triple (V, H, a(u, v» has a discrete spectrum from the theory in §2.6 and the condition u(l) = ° requires that if An is an eigenvalue, we have:

(2.454)

Let II du dv a l (u, v) = 0 dx dx dx .

For 4q2 - 1 ~ 0, we have

(2.455) II 4q2 - 1 2 a(u, u) = a1 (u, u) + 0 4x2 U dx ~ a1(u, u).

Thus, from Proposition 30,

(2.456)

d2 the problem relative to a 1 (u, u) being the Dirichlet problem for - dx2 on JO, 1[.

\08 Chapter VIII. Spectral Theory

We now seek a majorisation of An' To do this we evaluate the minimum of a(u, u) over those functions in V with lui = 1 and which vanish for x E [0, e], 0 < e < 1 (e given). For such a function:

II 4q2 - 1 fl 4q2 - 1 4q2 ----=-luI 2 dx = 2 u2 dx ~ 2

o 4x2 < 4x 4e

On noting that the eigenvalues of the operator - d 22 for the Dirichlet problem on

dx n2n2

]e, 1 [ are (1 _ e)2 ' n E N* we have from the Min-Max formula:

(2.457) A n2 n2 4q 2 - 1 [* * n ~ (1 _ e)2 + 4e2 ,V e E]O, 1 ,n EN, q EN.

Then taking e = ~ we deduce from (2.457):

I. An 1m 22 ~ 1,

n-oo n n

which together with (2.456) shows that for sufficiently large n

(2.458)

Hence for q E N* the zeroes fJ.q,n of Jq are asymptotically equal to nn. In the case q = 0 we will arrive at an analogous result either by proceeding directly or by making use of the recurrence relation:

(2.459) J q + I = - J q _ I + 2xq J q for q

and the result already obtained for q E N*.

9.4. Remarks on the Zeroes of the Eigenfunctions of the Laplacian

Let Q be a bounded open set in [Rn.

We denote by AI" .. , Ab ... (resp. u l , ... , Uk' •• ) the eigenvalues81 (resp. the corresponding eigenvectors) of the Dirichlet problem relative to the operator - .1, that is to say, the solutions in L2(Q) (or U(Q), 1 ~ P < 00) of the problem:

(2.460) {-LlU = Au in Q

ulaQ = O.

In this section we shall be interested in the zeroes of eigenfunctions Uk in Q. We start by giving the

Definition 10. Let v be afunction of class C(j2 on Q (v E C(j2(Q».

81 Counted with their multiplicities.


A connected open set (!) c Q such that:

(2.461 ) {i) v(x) # 0, Vx E (!)

ii) v E Cfjo (@) with v = 0 on o(!).

will be called a nodal domain of v.

We shall subsequently need to make use of the following lemma:

Lemma 8. Let v E Cfj2((!))nCfj1(@)(with (!) open in \Rn )

v > 0 on (!), v = 0 on o(!), (-Llvt E L1((!)) ,B2

Then vLlv E L1((!)), grad v E (L2((!))t and

(2.462) r Igradvl 2dx = - r vLlvdx J0 J0 (2.463) L (grad v) (dx = - L v grad (dx, V( Eel (@) .

Proof 1. From Sard's lemma,B3

a.e. kE]O,maxv[, (!)k = {XE(!); v(x»k}

is a regular open set (of class Cfj 2). Applying Green's formula,

-1 vLlvdx = 1 Igradvl 2dx -1, v ;~ dy. ~ ~ .~

But, v = k on O(!)k' whence

1 v OV dy = k 1 ~nV dy = k 1 Llvdx, a~ on a~V ~

then:

1 (Igrad Vl2 + (v - k)Llv)dx = O. e.

Now Igradvl 2 + (v - k)Llv ~ -(maxv)(-Llv)+ on (!)k

109

whence the result, in the limit as k ...... 0, after Lebesgue. 0

Proof 2. Given p E Cfj 1 (\R) with p = 0 in a neighbourhood of 0, we have p(v) E Cfj 1((!)) with compact support in (!). Thus

-1 p(v)Llvdx = 1 gradp(v). grad vdx = 1 p'(v) Igradvl 2dx.

82 (_ L1vj+ denotes the positive part of - L1v. 83 See, for example, de Rham [1]. Berger [1].

110 Chapter VIII, Spectral Theory

There exists Pk E '(5 1 (IR) with Pk = 0 in a neighbourhood of 0 and 0 ~ p~ ~ 2 and Pk(r) --+ r, p~(r) --+ 1, Vr > 0 ,

Pk(v)Llv + p~(v)/gradvI2 ~ -(2maxv)(-Llv)+

Pk(v)Llv + p~(v)/gradv/2 --+ vLlv + Igradv/ 2 ,

We can now state

Theorem 12. Let kEN * and t E IR such that t < Ak.

o

Let v E '(5 2( Q) such that ( - Llv)v ~ tv 2 in Q. Then the number of nodal domains of v is strictly less than k.

IJrooj Let N be the number of nodal domains of v. Suppose N ~ k, and let (1)1" .. , (1)k be the nodal domains of v with boundaries T1 , • •• , Tk • Put:

(2.464) Vj = {Ov III (1)j in Q\ (1)j .

The functions Vj are such that:

Put

{ VjEH1~), gradvj = (gradv)xeo/4

Vj E'(5(Q) , Vj = 0 on oQ.

Ek = {wEH 1 (Q)n'(5(Q),(w,udu = O,i = 1 to k - l,wlaQ = O},

where ui , i = 1, ... , k - 1, are the first k - 1 eigenfunctions of problem (2.460). der k

There exist constants ai' ... , ak, such that Fk = L ajvj belong to Ek and Fk t= O. j~ 1

I n effect there are k constants to choose with k - 1 constraints. Then:

(2.465)

from Corollary 2 (and Theorem 9), and Ek C H6(Q). On the other hand, from Green's formula (2.462):

(2.466)

Jl aJ L. vLlvdx. J

But by hypothesis:

-Lj

vLlvdx ~ t L Iv(xWdx = t In /v j(x)1 2 dx.

84 Where Xeoj denotes the characteristic function of the open set (!J j'

§3. Spectral Decomposition of a Self-Adjoint Operator 111

Thus

(2.467) itl L 1 ~:: 12 dx ~ t L IFk(xWdx;

hence (2.465) and (2.467) contradict the hypothesis t < Ak. o If Ak < Ak+ 1 and t = Ak the preceding theorem shows that N ~ k. The result which follows implies that this remains true if Ak = Ak + 1 .

Theorem 13. Let k EN* and v E~2(Q) such that (-Jv)v ~ AkV2 in Q. Then the number N of nodal domains of v is less than or equal to k.

Proof Suppose that N > k and choose Fk as in the preceding proof. Since (2.467) holds with t = Ab we have equality in (2.465), i.e.:

.f r laaFk 12 dx = Ak r IFk(xWdx; ,= 1 JQ Xi JQ

from Corollary 2 (to Theorem 9), Fk is an eigenfunction. The theorem will then result from the following lemma (see Chap. V).

Lemma 9. No eigerifunction of the operator - J for the Dirichlet problem can vanish on an open subset of Q unless it vanishes identically in Q.

In effect Fk is null in (Dk+l' but not in Q, which is a contradiction. o

Corollary 4. The number of nodal domains of an eigenfunction relative to the k-th eigenvalue of the operator - J for the Dirichlet problem in a bounded open set Q is less than or equal to k. In particular, if w is an eigerifunction relative to the first eigenvalue, then w(x) is different from zero for all X in Q. 0

§3. Spectral Decomposition of a Bounded or Unbounded Self-Adjoint Operator

Introduction

We shall first briefly recall the results obtained in §2 concerning the spectral decomposition of compact hermitian operators in a Hilbert space. Let H be a separable complex Hilbert space [the scalar product (resp. the norm) in H being denoted by ( , ) (resp. I I)], We have seen that if A is a compact hermitian operator in H, it possesses an at most denumerable system { VdH 1 of pairwise orthogonal eigenspaces Vk [Vk corresponding to the eigenvalue Ak E ~, dim Vk < + 00].

The subspace Va' possibly reduced to {O}, orthogonal to all the Vk is the kernel of A, and is not necessarily of finite dimension (but is separable if His).


On admitting Vo as an eigenspace corresponding to the eigenvalue AO = 0, we have:

(3.1)

with

(3.2)

Hence we have decomposed the space H into a direct sum of subspaces Vk in which A is reduced to multiplication by Ak •

If Pk denotes the orthogonal projection operator onto Vk , we can write

(3.3) {I = L. Pk

kEfIJ

Au = L. AkPkU, U E H . kEfIJ

The eigenvalues Ak belong to a bounded interval (a, f3) in the real line and have 0 as a point of accumulation. In (3.3) they appear 'disordered' relative to the order structure in IR. To get round this inconvenience, let us introduce

(3.4) {

f~r all A E IR

1) G" = EB Vk A.k::% A.

ii) E" the orthogonal projection onto G" ,

It is easy to see that the function A 1--+ E" has values in .!f(H) which satisfy the following properties:

i) E".EJl = Einf(".Jl)

ii) for all A, EH 0 = E" (3.5) [i.e. for all x E H , E H. X -+ E" x III H for e -+ 0]

iii) lim E" = 0 , lim E" = I . ).--00 A-+oo

In fact, since a(A) c (a, f3) c IR, this alone is enough to give E" = 0 for A < a,

and E" = I for A ~ f3. Moreover, the discontinuities of the function X 1--+ E" are the eigenvalues Ak , the jump in passing an eigenvalue being given by:

(3.6)

Thus, in the sense of distributions in IR with values in .!f(H),85 the derivative of E" can be identified with a measure denoted by dE" and given by:

(3.7) dE). = L. b "k @ Pk ,

"k "'" 85 See the vectorial distributions in Chap. XVI.


where b;'k = b(A - Ak ) is the Dirac measure at the point Ak • Hence, making use of the Stieltjes integral relative to the vector measure defined by E;., (3.3) can be written:

(3.8)

f+oo ii) A = _ 00 Ad E;. .

In this §3 we propose to show how the formulas (3.8) can be extended to the case of any self-adjoint operator A.

1. Spectral Family and Resolution of the Identity. Properties

Let H be a separable complex Hilbert space with the notations for scalar product and norm as used previously in the introduction.

Definition 1. A family {E;.} ;'E~ of orthogonal projections in H is called a spectral family or else a resolution of the identity if it satisfies these conditions:

(3.9)

i) El.E,.. = E inf (;..,..) A,jlEIR

ii) E _ 00 = 0, E + 00 = 1 ,

whereE_oox = lim Elx,E+oox = lim Elx forall xEH; A,-+oo

iii) EH 0 = E;., where EHOx = lim EH£x for all x E H .

The limits are taken in the norm of H.

£ > 0 £~o

Proposition 1. Let {E;'};'E~ be a spectral family; then,Jor all x, YEH, the function

(3.10) A ....... (E;.x, y)

is a function of bounded variation86 with total variation V (A; x, y) satisfying

(3.11 ) V{A; x, y) ~ Ixl.lyl, Vx, y E H, VA E IR.

Proof Let A1 < A2 < ... < An'

86 On every finite interval. (This will be understood throughout the following.)


def From (3.9) i) Ela,PI = Ep - Ea is an orthogonal projection.S ? From the

Cauchy-Schwarz inequality we have:

n n

L I(EIAj_l,AjIX, y)1 = I I(EIAj_l,).jIX, EI).j-l,Ajly)1 j=Z j=Z

n

~ L IEIAj_l,AjIxl.IEp.j_l,.l.jIYI j=Z

~ ( .t IEI.l.j_l,.l.jIXIZ)l/Z ( .t IEIAj_l,AjIYIZ)l/Z )-2 )-2

=(IEI.l.l,AnIXI2)1/2.(IEI.l.1').nIYIZ)1/2~ Ixl·lyl,

since (3.9) i) implies that EIAj _ 1, Aj I' Ep.j _ 1,.l.d = 0, i # j and for m > n m-l

(3.12) Ixl2 ~ IEI.l.",.l.mIXlz = I IEp.i . .l.i+dxlz. i;;::::;n

o

Corollary 1. Let {E.l.} AE~ be a spectral family. Then,for all A E IR, there exist the operators

EA+O = lim E", E).-o = lim E" ,,~.l.+0 ,,~).-o

Proof From (3.12), we see that if An -+ A - 0, then

lim IEIAj, Aklxz = ° ; j.k-OC)

in the same way (3.9) iii) is true for An -+ A + ° . o Proposition 2. Let f be a continuous function on IR with complex values and let x E H. Then it is possible to define for a < p, [a, P] c IR the integral

fl f(A)dE.l.x

as the strong limit in H of the Riemann sum:

{ If(Aj)(E).j+l - E.l.)x where a

(3.13) j

Aj E(Ai, Ai+ d

when max IAj + 1 - Ajl -+ 0. j

p

87 Recall that PE 2'(H) is an orthogonal projection if and only if p 2 = I and P* = P (see Chap. VI). Here, on the one hand we have

EJ •. pJEJ.,pJ = Ep - E.Ep - EpE. + E. = Ep - E. = EJ',PJ '

and on the other Et',PJ = E; - E: = Ep - E. = EJ.,Pl·

Note that we have: E[.,PJ = Ep - E.-o, E1.} = E. - E.- o, EJ-"',PJ = Ep, EJ-m,p! = Ep- o·


Proof The function f is uniformly continuous on the compact interval [a, P]. Hence for all I: > 0, 36 > ° such that

(3.14) I). - ,1,'1 < 6 => If(A) - f(,1,')1 < I: .

Now consider two partitions of [a, P]:

and let

IX = A1 < A2 < ... < An = p, max I Aj+ 1 - Aj I < 6 j

a = 111 < 112 < . . . < 11m = p, max I 11 j + 1 - 11 j I < 6 j

a = V1 < V2 < ... < vp = p, p ~ n + m

be the partition resulting from the previous two. Then if 11~ E] 11k' 11k+ 1]' we have

Then the norm of the second member of (3.15) satisfies

I ~ I:sEJvs.vs+d X 12 ~ 1:2 1 ~ E]vs,vs+d X 12 = 1:2lEja,lljxl2 ~ 1:21x12. 0

Definition 2. For any given XEH and any continuous function f on IR, the integral

f+OO III -00 f(A)dE"x is defined as the strong limit in H, if it exists, of a f(A)dE"x when

a -+ - 00 and p -+ + 00.

We then have:

Theorem 1. For x given in Hand f a complex valued continuous function on IR the following conditions are equivalent:

(3.16) f-+oooo f(A)dE"x exists;

(3.17) f-+oooo If(A)1 2dIE"xI 2 < + 00 ;

deC f + 00

(3.18) y f--+ F(y) = -00 f(A)d(E"y, x) is a continuous linear form.

Proof 1st step: (3.16) implies (3.18).

r:ffif7~)::.:t:st:a:~:::::;n::::::~O:: ;ith a Ri,mann sum appmximating Hence from the fact that E" is an hermitian operator (orthogonal projection) we


have (y, E;.x) = (E;.y, x) and in virtue of the Uniform Boundedness theorem,BB we get (3.18). 2nd step: (3.18) implies (3.17).

(P-Put y = Ja f().) dE;.x; then by applying the operator E]a.p] to a Riemann sum

approximating the integral that defines y and by using (3.9) i), we see that

y = E]a,p]Y· Hence again from (3.9) i):

f+OO (P' F(y) = _ 00 f().) d(E"x, y) = a'~~ 00 Ja' f().) d(E"x, y)

= lim a'--+ - 00

P'~+oo

= lim a'-+-oo P'~ + 00

P'~+oo

we deduce that I y 12 ~ II F II .1 y I whence I y I ~ II F II. Furthermore, approximating (P-

Y = Ja f().) dE"x by Riemann sums and again using (3,9) i) we get:

lyl2 = 1 r f().)dE;.x 12 = r If()'WdIE"xI2 ,

so that

IP If()'WdIE"xI 2 ~ IIFI12;

hence letting IX -+ - 00 and 13 -+ + 00, the result (3.17) follows. 3rd step: (3.17) implies (3.16). For IX' < IX < 13 < 13', we have:

11~' f()')dE"x - r f()')dE"XI2 = f If()'WdIE"xI 2 + J:' If()'WdIE"xl

whence the result. 0

Theorem 2. Let). 1-+ f().) be a real-valued continuous function. Let D(D c H) be defined by:

(3.19) D = {x E H; I-+oooo If()'WdIE"xI 2 <; + oo} .

Then D is dense in H and a self-adjoint operator T is defined in H by:

(3.20) (Tx, y) = I-+oooo f()')d(E"x, y), Vx E D, y E H ,

88 See Chap. VI, § I.


with domain D(T) = D. We have

(3.21) TEA ::::J EAT (that is, TEA is an extension of EA T) .

Proof F or all y E H and for all c > 0, there exists ct, /3, - 00 < ct < /3 < + 00

such that Iy - E1a.fJ1yl < c. We further have

t+oooo If(AWdIEA.Ela.fJlyI2 = r If(AWdIEAyI2;

E1a.fJ1y ED and from (3.9) ii) i5 = H. Since f(A) = f(A.), (EAX, y) = (EAy, x), Tis symmetric.89

Let y E D(T*), put T*y = y*; then since E1a.fJ1z ED, \/zEH, we have:

(z, E1a.fJ1y*) = (E1a.fJ1z, y*) = (E1a.fJ1z, T*y) = (TE1a.fJ1z, y) = r f(A)d(EAz, y)

whence we deduce that

and from Theorem 1, t+oooo If(A)1 2dIE AyI2 < + 00 whence y E D. Thus

D = D(T) ;2 D(T*). Since T is symmetric, we have T ~ T*, hence T = T *. Finally, let x E D( T); applying the operator E,.. to an approximation of the integral

Tx = t+: f(A) dEAx, and using (3.9) i) we get

E,..Tx = t+: f(A)d(E,..EAx) = t+oooo f(A)d(EAE,..x) = TE,..x. 0

Corollary 2. In the particular case where f(A.) = A, we deduce that

Y E H

(3.22)

We then write symbolically

(3.23) A = t+oooo AdEA

and call (3.23) the spectral representation of the self-adjoint operator A in the Hilbert space H.

89 See Chap. VI, §3.


Corollary 3. For T = f-+: f(A) dE;. given by (3.20), we have

(3.24) I Txl2 = f:~ If(AW dIE;.xI2, x E D(T) .

In particular if T E !l'(H),90.91 then

(3.25) (T"x, y) = f:~ [f(A)]"d(E;.x, y) for x, y E H, n E N* .

Proof Verification of(3.24). Since E;.Tx = TE;.x for x E D(T), we have

(Tx, Tx) = tf(A)d(E;.X, Tx) = tf(A)d(TE;.X, x)

= tf(A)d;. {{f(Jl)d"(E,,E;.X, X)}

= t f(A.)d;. {roo f(Jl)d"(E,,x, x) r2

= t If(AW dlE;.xl2 •

Verification of(3.25): This is analogous to that of (3.24). o We will now show that in the particular case where f(A) = (( - A)-I, the operator T so obtained is the resolvant R(() = (0 - A) -1 of the operator A.

Theorem 3. Let {E;'};'E Oil be a spectral family in the Hilbert space H. For ( E C - ~ and x, y E H, we put:

(3.26) f+ 00 1 1((, x, y) = _ 00 ( _ A d(E;,x, y) .

i) Then for (E C - ~, there exists a unique R (() E !l' (H) such that

(3.37) 1((, x, y) = (R(Ox, y) for all x, y E H ;

ii) R(() is the resolvant associated with the self-adjoint operator A defined in Corollary 2.

Proof 1) Point i). Let x be fixed in H. The mapping y 1--+ 1((, x, y) is ~ntilinear in y.

90 That is if T is bounded. The result is equally true when T is not bounded with the customary precautions over the domains. 91 Indeed if f is bounded then T is bounded. 92 From (3.9) i).


Moreover, since the total variation v(2, x, y) of the function 2 f--+ (ElX, y) is < Ix I ·1 y I,

1 1 11(" x, y)1 < Ixl.lyl ~~~ 12 _ 'I < Ixl.lyllIm 'I .

From the Riesz Representation Theorem, there exists R(,) E !f (H) such that

I(C x, y) = (R(,)x, y) for all x, y E H .

2) Point ii). Note that

fl 1 = ~ d (EJl x, y) ,

-00" }l and

IEl R(,)xI 2 = foo I' ~2 }l12 d(EJlx, x),


1-+0000

22 dIE l R(,)xI 2 = 1-+0000 I' ~2 212 d(E;.x, x) < + 00 ,

so that Rmx E D(A) defined by (3.22). Furthermore, a simple calculation shows that

f + 00 1 (AR(,)x, y) = -00' _ 2 d(E;.x, y),

whence we deduce that

((0 - A)R(,)x, y) = (x, y) for all x, y E H .

On the other hand, if x E D(A), then we again verify that

(R(O(O - A)x, y) = (x, y) for all x, y E H

and R(,) coincides with the resolvant of A for' E IC - IR and thus everywhere this resolvant is defined. In conclusion of our studies in this Sect. §3.l we have arrived at

Theorem 4. i) There exists an injective mapping 8:93

(3.28)

from the set of spectral families in the Hilbert space H into the set of self-adjoint operators on H.

93 See for example Kato [1]. p. 358 for the injectivity of this mapping.


ii) Let A be the self-adjoint operator associated with a spectral family { E;.} ;'e ~ (under the mapping 8); we have:

(3.29) D(An) = {x E H; f-+oooo A2n d(E;.x, x) < + oo}, n ~ 1

and for x E D(An), y E H

(3.30)

f+oo (Anx, y) = -00 And(E;.x, y)

iii) If Pn(A) is a complex polynomial of degree n, then we define Pn(A) by:

(3.31) (Pn(A)x, y) = f-+oooo Pn(A)d(E;.x, y), X E D(An) , y E H

iv) If f is a continuous complex function on IR, we can put:

(3.32) f(A) = T where T is defined by (3.20) .94

Proof This follows immediately from the preceding theorems. o In the nex.t Sect. §3.2 we go on to show that the mapping 8 in Theorem 4 is bijective.

2. Spectral Family Associated with a Self-Adjoint Operator; Spectral Theorem

Let A be a self-adjoint operator in a separable complex Hilbert space H. In this section we use the notation R(() = - R(() = (A - (I)-I. We first establish the following result:

Theorem 5.

(3.33)

For all x, y E H there exists a function of bounded variation

{ t f-+ p(t; x, y)

oftotalvariationv(p) ~ Ixl.lyl

such that for all ( E C - IR we have the representation:

(3.34)

94 Note that P.(A) and f(A) are then normal operators.

§3. Spectral Decomposition of a Self-Adjoint Operator

this function p is unique if we impose the conditions (normalisation conditions):

(3.35) for all x, Y E H, p( - 00; x, y) = lim p(O; x, y) = 0 ; (J-+-oo

(3.36) for all t E IR, p(t + 0; x, y) = p(t; x, y) .

Proof Let x E H. Put:

(3.37) q>AO = (R«)x, x) for ,d:: - IR .

The function q>x is analytic in C - IR (see Proposition 2 in § 1). Recall that R(O satisfies

(3.38) {R~O - R(C~ ~ (' - C) R(O. R(C) [R(O]* = R(O,

so that, on putting' = ~ + i'1, the imaginary part of q>x satisfies

1m (r) = q>x (0 - q>AO q>x ." 2i

(R(Ox, x) - (R(Ox, x)

2i

-(R(Ox, x) - (R([)x, x)

2i

thus, from (3.38)

«R(O - R([))x, x) = (' - [)(R(OR(nx, x) = 2i'1IR([)xI 2 ,

so that

(3.39) {' = ~ + i'1 1m q>x(O = '1IR(OxI 2 > 0, if '1 > 0, x '# O.

Moreover, from Proposition 2 in § 1, we have:

- Ixl IR(Oxl ~-,

'1 whence

(3.40) sup {'1Iq>x(i'1)I} ~ Ixl2 . ~ > 0

121

Thus95 if the function , ~ q>( 0 is holomorphic in the half-plane 1m' > 0, with its

imaginary part ~ 0 in this half-plane and such that sup {'1Iq>(i'1)I} < + 00, then ~ > 0

it admits the representation:

(3.41) q>(O = f+oo dw(t) (= ~ + i'1, '1 > 0 -00 t - , '

95 See Akhiezer-Glazman [1].


where t H w (t) is an increasing function of bounded variation in the wide sense, which is unique if we impose the conditions:

(3.42) { w(-oo) ,= a~~oo w(tx) = ° w(t + 0) = w(t) (right continuity) ;

Thus for 1m ( > ° we have

(3.43) - f+oo dw(t; x) cpAO = (R(Ox, x) = -00 t _ ( .

We also have

and

(3.44) (R([)x, x) = f+oo dw(t; x) . -00 t - (

Since, whenever ( is in the half-plane 1m ( > 0, [is in the half-plane 1m ( < 0, we have the representation:

(3.45) {for all (E if ~+: an~ for all

(R(Ox, x) = f dw(t, x) . -00 t - (

X E H

Now for x, y E H, we define p(t; x, y) by

(3.46) 1

p(t; x, y) = 4 [w(t; x + y) - w(t; x - y)]

i + 4[w(t; x + iy) - w(t; x -iy)].

Then we have:

(3.47) (R(Ox,y) = f+oodP(t~X,Y), x,YEH, (Eif -IR. -00 t

We note that t H p(t; x, y) is for each x, y, a complex valued function of bounded variation satisfying

(3.48) {lim p(tx; x, y) = p( - co; x, y) = ° for x, Y E H

;;t -+00 0; x, y) = p(t; x, y) (right continuity) .


It is easily verified that the representation (3.47)-(3.48) is unique: in effect if this is not the case, then there will exist a complex function of bounded variation a(t) = cx(t) + ifJ(t) such that

f+OO da(t) = f+oo da(t)_ = 0, Vz E IC - IR; -oot-z -oot-z

then also

f+OO da(t) --=0,

-00 t - z

which implies

f+OO dcx(t) = f+oo dfJ(t) = 0, Vz E IC - IR; _oot-z _oot-z

from (3.41)-(3.42) it follows that cx(t) = fJ(t) = 0. From the uniqueness of the representation [(3.47)-(3.48)] we get

(3.49) {i) p(t; x, y) = p(t; y, x)

ii) x ~ p(t; x, y) is linear.

Thus

(3.50) for all t, p(t; x, y) is an hermitian sesquilinear form.

We note that from the uniqueness of the representation of cpAO, we have:

{for all t E IR ,

(3.51) p(t; x, x) = (O(t; x) ~ 0 if x # 0,

t ~ p(t; x, x) being increasing in the wide sense.

From the Cauchy-Schwarz inequality

(3.52) { / p(t; x, yW ~ p(t; x, x)p(t; y, y)

~ p(+ 00; x, x)p(+ 00; y, y) for all t E IR,

so that it is sufficient to establish

(3.53) p(+ 00; x, x) ~ /X/2 for all x E H

in order to complete the proof of Theorem 5.

For IX > 0, we have for '1 > 0

I f+Gt '1 dp(t; x: x) I ~ If+ oo '1 dP (t;x:X)1 + 11 + 12 ,

- • t - 1'1 _ 00 t - 1'1

with

11 = f-·I ~ I dp(t; x, x) ~ f-· dp(t; x, x), - 00 t 1'1 - 00


and

1+00 I 11 I 1+ 00 12 = ~ dp(t; x, x) ~ dp(t; x, x) ,

~ t 111 ~

so that

I f +~ dp(t; x, x) I -11 . ~ I 11 (R(i11)X, x)1 + p( -a; x, x) + p( + 00; x, x) - p(a; x, x) .

-~ t - 111

Since 111(R(i11)X, x)1 ~ Ix1 2, we have:

(3.54)

11 ': ~ Ixl2 +p(-a; x, x) +p(+oo; x, x) -p(a; x, x). I f +~ dp(t· x x) I _~ t - 111

In (3.54), letting 11 -+ + 00 we get

(3.55) I L+~IX dp(t; x, x) I ~ I Xl2 + p( -a; x, x) + [p( + 00; x, x) - p(a; x, x)]

then letting a -+ + 00 in (3.55) we get (3.53). 0

Theorem 6. With the hypotheses of Theorem 5, there exists a spectral family {E).}).elR such that

(3.56) p(A.; x, y) = (E).x, y) for all x, y E H .

Proof From (3.52)-(3.53), the hermitian sesquilinear form p(A.; x, y) is continuous on H x H; by the Riesz Representation Theorem there exists an hermitian operator E). E .5l'(H) satisfying (3.56). We thus have

(3.57) (R(Ox, y) = L+: d~E~,t for all x, y E H, CEC - IR.

We now show that {E).} is a spectral family. From (3.57) we deduce that

{ (R(Ox, R((i)y) = f:,oo•

OO

t ~ C d(Erx, R((i)y) (3.58)

for all x, y E H, C, C E C - IR .

Furthermore,

- -- - - 1 - -(R(Ox, R(C/)y) = (R(nR(Ox, y) = " _ C {(R(C)x, y) - (R(Ox, y)}


Hence we also have the representation

f +oo 1 { fr 1 } (R(Ox, R«(')y) = -00 t _ ( d r -00 S _ (' d(Esx, y)

which must coincide with the representation (3.58). Thus:

fr 1 - - -~-, d(Esx, y) = (Erx, R(ny) = (R(nErx, y) ,

-00 s - (

and

f r 1 f+oo 1 ~ d(Esx, y) = ~ d(EsErx, y),

- 00 s \, - 00 S \,

and since these representations are identical, we have

(3.59)

thus we have:

which is (3.9) i).

{ (EsX' y) = (EsErx, y), for all x, y E H ;

s ~ t,

From this it follows in particular that

(3.59)' Ei = E;. thus E;. is a projection.

125

Since E;. is hermitian, it follows that it is an orthogonal projection. From (3.35)-(3.36)96 we immediately have

It remains to show that lim Er = I. 1- + 00

The function t 1-+ 1 Er x 12 = (Erx, x) is for each x E H an increasing function of t majorised by Ix1 2, whence we deduce that lim Er = Eoo exists .97

t- + 00

Put F = I - Eoo. We have

FEr = Er - EooEr = Er - lim EsEr = Er - E; s- + 00

Then for all x, YEll,

f+oo 1

(R«()Fx, y) = ~ d(EJx, y) = 0 , - 00 t \,

96 And also from (3.59) and (3.59)' those following equalities hold in the sense of (3.9) ii) and (3.9) iii), that is to say, in the strong sense. 97 In the strong sense, i.e. "txEH, lim E,x = Ecnx.

t- + 00


which implies that for all x E H, R(()Fx = 0, whence for all x E H, Fx = 0 and F = o. Hence (3.9) holds and Theorem 6 is proved. o Corollary 4. For all Il E IR and for all ( E IC - IR, we have

(3.60)

Proof

f+OO 1 (EAR(Ox, y) = (R(Ox, EAy) = - --=--r d(Erx, EAy)

- 00 t ~

fA 1 - --=--r d(Erx, y) .

- 00 t ~

On the other hand,

f+OO 1 - f" d(Erx, y) (R(OE;x, y) = - --=--r d(ErEAx, y) =

-00 t ~ -00 t - ( o

Remark 1. i) Evidently we have:

(3.61 )

as is immediately verified. ii) If 8 denotes the mapping defined in Theorem 4 and {E,,} "e IR the spectral family associated with A by Theorems 5 and 6, then:

(3.62) o We then have

Theorem 7 (Spectral Theorem). Let H be separable complex Hilbert space. i) There exists a bijective mapping 8 from the set of spectral families in H onto the set of self-adjoint operators on H. ii) Let {E,,) J.E IR be a spectral family and A = 8( {E;.) J.E IR). Then E A and the resolvant R(O = ((I - A)-l are connected by

{ f+OO I (R(Ox, y) = _ 00 ( _ Il d(E"x, y)

for all x, y E H, ( E IC - IR .

(3.63)

and

(3.64) {

II "2 [(E"x, y) + (EA - ox, y)] - "2 [(El'x, y) + (EI' - ox, y)]

= lim _1_. f (R(Ox, y)d(, for J.i < Il , e ~ 0 + 27[/ r,

98 The symbol =0 signifies here an extension of the operator (see Chap. VI).


r. being the contour shown below:

ilR

II + it A + it I I I I IR

A

II - it: A - it

r-, Fig. I

Proof. The points i) and (3.63) follow from Theorems 4 to 6. To demonstrate (3.64) put: p(t) = (E,x, y) and let

(3.65) f +OO 1 f+oo 1 1(0 = -00 t _ ( dp(t) = -00 (t _ 02 p(t)dt.

using integration by parts. It is easily found 99 that

-21 . f 1(Od( = -~ f+oo 1 2 {p(A + et) - p(p, + et)}dt. 1rl r, 1t _ 00 1 + t

Then using the Lebesgue dominated convergence theorem and the right continuity of the function t ~ p(t), we get

lim {-~f I(Od(} = ~([p(A - 0) - p(p, - 0)] [Arctgtnoo • ~ 0 21tl r, 1t

+ [p(A) - p(p,)][Arctgt]tOO},

which is (3.64). o In conclusion, we have

Corollary 5. Let H be a Hilbert space and A a self-adjoint operator on H; then there exists a spectralJamily {E.d.icd! such that

(Ax, y) = f n;l Ad(E"x, y) ,

Ax = f n;l Ad(E"x) .

99 On taking r, to be the two straight line segments parallel to the real axis [Jl - ie, .l. - ie] and [.l. + ie, Jl + ieJ.


We use the symbolic notation

3. Properties of the Spectrum of a Self-Adjoint Operator. Multiplicity. Examples

3.1. Properties of the Spectrum

Theorem 8. Let A be a selfadjoint operator on a separable complex Hilbert space H. Let:

a(A) = ap(A) u ac(A) u a,(A) be the spectrum of A ,

{E,.}AEIR the spectral family of A .

Then we have the following properties: i) a(A) c IR;

ii) Ao E ap(A) (point spectrum of A) if and only if Elo i= Elo - O; the corresponding eigensubspace is then v;'o = Plo(H) where

Plo = Elo - Elo - O ;

iii) Ao E ac(A) (continuous spectrum of A) if and only if we have:

(3.66) { i) Elo = Elo - O ii) 'If, > 0, Elo -< i= Elo +<;

iv) the residual spectrum a,(A) is empty.

Proof 1st) Point i) is given by Proposition 2 in §1.

f+OO

2nd) We have A - AoI = -00 (.A. - Ao)dE l , and for all x E D(A)

(3.67) f+OO

I(A - AoI)xl2 = -00 (.A. - Ao)2dIE;.xI2 .

Suppose that Ao is such that E lo i= E lo _ 0, then there exists y E H with (Elo - Elo-O)y = x i= 0 and x E D(A) and (3.67) holds. ButlEl xI 2 = (El(Elo - ElO-O)Y' x) is equal to zero ifA < Aoandisindependent of A if A > Ao. Hence diEAxl 2 = 0 for A i= Ao and (3.67) implies that Ax = AoX.

100 It should be noted that whilst every self-adjoint operator is characterised by its spectral family {EJ i.E~' it is not, in contrast, characterised by its spectrum cr(A).


Conversely, if x is an eigenvector of A for the eigenvalue )-0, (3.67) implies that

o = f~coco (A - Ao)2dIE).xW ~ [;2 f dlE).xl2 for all [; > O. 1..1 - ..101 :;, e

We then deduce that

IE+oo(xW = IE).o+'(xW

IE-co(xW = IE).o-'(xW .

Taking note of E+oo(x) = x, E_oox = 0 and of the right continuity of A f-+ IE).xI2, we get:

whence follows

and

whence ii).

{ lxl2 = IE).o+oxI2 = IE).ox12 = (EAox,x)

o = IE).0_oxI 2 ,

3rd) We now prove that the residual spectrum is empty. If Ao E 0",( A) then i) implies that Ao E IR and since the closure of D((A - Aol)-l) #- H there exists y #- 0, Y E H such that ((A - Ao)X, y) = 0 for all x E D(A). Then:

(Ax, y) = Ao(X, y) = (x, AoY) , Vx E D(A) ,

whence we deduce that y E D(A*) = D(A) and Ay = AoY and thus y E O",(A) n O"p(A) which is a contradiction. Consequently O",(A) = 0 whence iv). 4th) Suppose that Ao ~ O"(A). Then the resolvant UoI - A)-l exists and is continuous. Hence A).o = AoI - A has a continuous inverse which means that there is a constant IX > 0 such that

I(A - Ao)xl ~ IXlxl for all x E D(A),

Suppose that Ao be a point such that E). = E). -0, but E). __ , #- E). +t for all o 0 0 0

[; > 0 sufficiently small (such a point is called a continuity point of the spectrum but it is not a point of constancylOl of the spectrum). Choose [; such that 0 < B < IX.

101 Ao is a point of constancy of the spectrum if E,o = E,o-o. E, = E,o in a neighbourhood of Ao.


Since E;.o-' #- E;.o+E' there exists Y E H with x = (E;.o+' - E;.o-')Y #- 0, x E D(A). Applying (3.68) to x so defined, we get:

whence

A > Ao + e

Ao - e ~ A ~ Ao + e ,

A ~ Ao - e

which, because of the way e has been chosen, is a contradiction. It is further easily verified that if Ao is a point of constancy of the spectrum then there exists a constant (X for which (3.68) holds, which implies that Ao E p(A). We have thus established that if Ao satisfies (3.66) then Ao If p( A) and Ao If 0' p( A); bearing in mind that O'r(A) = 0, we see that Ao E O'c(A) and conversely. 0

Theorem 9. Let A be a bounded self-adjoint operator l02 in H. Then

(3.69)

We write

inf (Ax, x) Ixl = I

{ i) ;'}~[A) A =

ii) sup A = sup (Ax, x) . ;'E<1(A) Ixl = 1

def (Xl inf (Ax, x)

Ixl = I

def (X2 = sup (Ax, x) .

Ixl = 1

Proof Let Ao E O'(A). For all e > 0, we can, from Theorem 8, find Y. #- 0 such that:

(E;.o+' - E;.o-')Y' = Y.

and we can assume IY.I 1. Then IE).Y.12 = IE).(E). +. - E;. _.)y.1 2 , and o 0

(AYt' Y.) = f)'o+t Adl(E;. - E;'0_.)y.1 2 ; A.o -e

letting e tend to 0 we deduce that

Ao lim (AyE' Ye) , e ~ 0

102 Thus hermitian.



(3.70) sup A = sup Ao ~ IX2 . lea(A)

Now suppose that IX2 ¢; a(A). From Theorem 8 we can find Al and Az with

Al < IX2 < A2 and El , = E;'2 .

Then since

131

at least one of the operators I - E;'2 and E;" is t= O. If I - E;'2 t= 0, then we can find y (Iyl = 1) such that (/ - El,)y = y. In this case

(Ay, y) = f:: AdIE;,(/ - El2 )y12 = f:2'" AdiElyl2 ~ A2 > IX2 ,

which is absurd. Thus necessarily we have El = E;, = I. But then for all Z E H, 2 ,

with Izi = 1, El,z = Z and we get:

(Az,z) ~ Al < IX 2 ·

Hence the hypothesis IX2 ¢; a( A) is absurd and we have thus proved (3.69) ii). An analogous argument l03 proves (3.69) i), whence Theorem 9 is established. 0

The following Krylov-Weinstein Theorem (see Yosida [1], p. 321) gives an approximation to the spectrum of a self-adjoint operator A.

Theorem 10. Let A be a self-adjoint operator in H. For all x E D(A) such that Ixl = 1, we define:

(3.71) IXx = (Ax, x), f3x = IAxl . Then for all e > 0, we can find At E a( A) satisfying

(3.72) IXx - (f3~ - IX~)I/Z - e ~ At ~ IXx + (f3~ - IX~)1/2 + e.

Proof Note that we have:

whence:

103 With A replaced by - A.


Then if IEAXl2 does not vary in the interval given by (3.72) we obtain

fJ~ - a~ ~ ((fJ~ - a~)1/2 + E)2 > fJ~ - a~

which is a contradiction. o Remark 2. The Rayleigh Principle consists of taking ax as an approximation to the spectrum of the operator A. If one calculates fJx then (3.72) gives an upper bound for the error in taking ax as an approximation of the spectrum of A. The quantity ax called the Rayleigh quotient is very useful in numerical methods for calculating eigenvalues. For concrete applications of these estimates, consult Yosida [1]. 0

Proposition 3. Suppose that A is a self-adjoint in H such that

(3.73) {there exists a constant a > 0 such that

(Ax, x) ~ alxl2 for all x E D(A) .

Then the spectral family {E A} .I ElKl of A is such that

(3.74) EA = 0 for A. < a.

Proof From (3.64) for A. < f.1 < a and x, y E H, we have

1 1 104 2 [(ElLx, y) + (Ell-OX, y)] - 2 [(EAX, y) + (EA-Ox, y)]

= lim - ~ f (R(Ox, y)d( ,~o+ 27r:1 r,

where the contour F, is defined in Theorem 7. But from (3.73) and Theorem 1 in §l, it follows that ( f-+ R( 0 is analytic in the halfplane Re ( < a, and F, for sufficiently small E is a closed contour in this half-plane. It follows that

lim - ~21 . f (R(Ox, y)d( = O. ,~o 7r1 r,

Since for A. -+ - 00, E A -+ 0, we have:

(ElLx, y) + (Ell-OX, y) = 0 for all x, y E H ,

whence we deduce that IEILxl2 = 0 for all x E H, f.1 < a. 105

3.2. Multiplicity of the Spectrum

o

We start by considering the case where H is a space of finite dimension, dim H = n. Now let A be a self-adjoint endomorphism of H and let

p ~ n

104 Note that necessarily we have E" = E"-o, E, = E,-o (A and /1 cannot be eigenvalues). 105 One can also remark that Proposition 3 follows directly from Theorem 9.


be the eigenvalues of A with multiplicities respectively equal to p

m1 , m2, ... , mp , L mj = n. j=l

133

Denote by V;'j the eigenspaces relative to the eigenvalues A.j and by x j " ••• , X jmj an orthonormal basis for V;. .

j

If {E;.} is the spectral family associated with A, then if we put P;'j = E;'j - E;.r o, we have

V;. = P;JH), J J

and the set {Xj.;s = l, ... ,mj , j = l, ... ,p}

is a complete orthonormal system for the finite dimensional space H. For all y E H, def

we have, with aj. = (y, Xj):

(3.75)

Now if a < P, (a and P E IR), we have:

(3.76)

and

(3.77)

Hence for a, P fixed such that a < P and for M a given subspace in H, the set {(Ep - E«)y; Y EM} does not contain V;. if dim M < mj' Further, for suitable M

• J

such that dim M = mj , the set

{(Ep - E«)y; Y E M}, a < A.j ::::;; P . contains V)..'

J

In particular, we will have:

(3.78) m1 = ... = mp = 1 , p = n

if and only if there exists a vector y E H such that the set {( E p - E«) y; a < P} generates the whole space H. Such considerations can be extended to the case where the Hilbert space H is infinite dimensional and' separable, with A self-adjoint and compact or self-adjoint with a compact resolvant. This leads to the following definitions:

Definition 3. The spectrum of a self-adjoint operator A on a Hilbert space H is said to be simple if there exists a fixed vector y E H such that the vector subspace generated by the set


(3.79)

is dense in H.

Definition 4. Let A be a self-adjoint operator on a Hilbert space H. For fixed ct < f3 we consider the vector subspaces M contained in (Ep - Eo)H, such that

(3.80) (Ep - Eo)H = {(El - E,..)M; J1. < A, (A, J1.) c (ct, f3)} .107

Then in! {dim M, M satisfying (3.80)} is called the total multiplicity of the spectrum of A in the interval] ct, f3].

Definition 5. The spectral multiplicity of the operator A at a point A = ,1.0 is defined as the limit as n --+ + 00 of the total multiplicity of the spectrum of A contained in the interval] ,1.0 - 1 In, ,1.0 + 1 In].

3.3. Examples

Example 1. Let H = L 2(1R) and A the operator of multiplication by t, defined by

(A!)(t) = tf(t) , fEH, t E IR ,

The operator A is self-adjoint with domain

(3.81) tfE U(IR)} .

Let

(3.82) Xl = {Xl(t) = 1 if t~,1.

Xl(t) = 0 t > ,1..

The operator A admits a spectral family {E l} defined by

(3.83) Ed=xl.f forall fEL2(1R);

we have:

(Ed,!) = f~C() If(tWdt

and

d(Ed,f) = If(,1.W d,1. a.e. A,

so that we then have

106 Where {EA} is the spectral family for A. 107 In (3.80), the { } denote the vector subspace generated by the (EA - E.)M, with the indicated conditions on ..l and /1.


The function A H E;. is continuous at each A E IR and has no spectral constant point so that

(3.84) O'(A) = O'AA) = IR .

Moreover, this continuous spectrum is simple. In effect, denote by g the scalar function defined by

(3.85) g( t) = Ck > 0, t E ] k - 1, k], k E 7l. ,

with

(3.86)

so that

(3.87)

Writing

(3.88) X~.P = Xp - X., (X, < p , (where the characteristic function X is defined by (3.82)), we easily see that the vector space generated by

(3.89)

is dense in the space of step functions with compact support and thus dense in L2(1R). 0

Example 2. Here we consider H = L 2 (IR) and the operator

(3.90)

with domain

(3.91 )

d B = i

dx

where HI (IR) is the Sobolev space of order 1 constructed on IR (it is plain that B is self-adjoint (08).

Denote by ff the Fourier transform in L 2 (IR).I 09

If A is the operator of multiplication by the variable x as defined in the previous example then we have:

{A=ff-1Bff

(3.92) B=ffAff- l ,

ff being an isometry on L2(1R) (i:e. a unitary operator); we say that A and Bare unitarily equivalent.

108 See Chap. VI.

109 See the Appendix "Distributions" in Vol. 2. Here we take :Ff(y) = -'- r e-iY'f(x)dx. jb.JIR


In particular it follows that the qualitative properties of the spectra of A and Bare the same. From Example 1, the spectral family {Et} associated with the operator A is multiplication by the function X;,. defined by (3.82). It is related to the spectral family {En associated with B through

(3.93) E~ = .? Et.? -1 .

Hence if f E U( /R), and E~p = E: - E~, rx < /3, then E:p = E: - E~ is defined by

(3.94)

and the spectrum of B is

(3.95)

This spectrum is simple.

Example 3. We now give an example of a multiple spectrum. First we note:

(3.96)

if { E;,.} is the spectral family associated with a self-adjoint operator A in H, then {D;.} with

D;. = EJ;. - E_j~ -0 = E[ _j~. + j~] (A ~ 0)

is the spectral family associated with the operator A 2 ,

the easy proof of which we leave to the reader. Now consider:

(3.97) { DJ(t) = ~ f+oo sinj}:(t - r) f(r)dr,

11: -00 t-r

fE L2(/R).

From (3.82), (3.93), (3.97), we have:

(3.98)

so that {D.<} is the spectral family {EX} associated with the operator C = B2 defined by d 2

(3.99) C = - -2' D(C) = H2(/R) dx

(H 2(/R) Sobolev space of order 2 constructed on /R). The spectrum of the positive operator C is spread along the half-line /R + = [0, + 00 [. We show that this spectrum is not simple. In the contrary case there would exist g E L2(/R) such that the subspace generated by the { D.<g } ;. ;. 0 is dense in L 2( /R). Then g would also be a "generator" element for


B, that is to say such that {E~llg; IX < fl}IIO is dense in L2(1R). Consequently, for o < IX < fl, there would exist (see §3.5, (3.142) below) cP = CPall E L2(IR,dO'), dO' = d(Dtg,g), such that:

(3.100) f+OO

E~llg = 0 cp(t)dDtg

and


(3.101) 2 { 1 if t E (IX, fl)

cp( t ) = . o If t E ( - fl, - IX)

which is impossible. In fact, the spectral multiplicity is 2 at each point. To see this we note that we can choose the generator element associated with B to be an even function gl (resp. an odd function g2)' Furthermore, every function I E L 2( IR) decomposes into the sum of two functions, one II even, the other 12 odd. Then there exists CP.i E L 2 (IR, dO'), with dO'j = d(E~gj' g), gj E L 2 (1R),j = 1,2, such that:

(3.102)

and

(3.103)

it is easily verified that

(3.l04)

Since .I} ( t) = (-1)j-I.I)( - t), j = 1,2, it follows from (3.l02) that

(3.105)

so that

(3.106)

d2 Hence {g I, g2} forms a system of generators for the operator C = - - and the

dx 2

spectral multiplicity is equal is 2.

110 This notation also denotes the vector subspace generated by the E:p9, IX < p.


o

4. Functions of a Self-Adjoint Operator

Let A be a self-adjoint operator in a Hilbert space H, A = fAdE;., its spectral

decomposition (see Theorem 7 and Corollary 5). We have already defined in §3.1 above (see Theorem 4): i) the integer powers of A:

A" = fIR A" dE;., i.e.:

(3.107) x E D(A") ¢> fIR A2n dlE;.,xl2 <

n = 1,2, ... ;

ii) the polynomials of A:

if P k is a polynomial of degree k, then

(3.108) Pk(A) = fIR Pk(A) dE;., , i.e.

+00

D(Pk(A)) = D(Ak) , fIR IPk(AW dlE;,(xW < + C1J, X E D(Ak) ;

iii) f (A) for a function f continuous on IR,

if f is such a function, then

(3.109) f(A) = fIR f(A) dE;.,

X E D(f(A)) ¢> fIR If(AWdIE;.,xI 2 < +00,

the notation is justified by the fact (for example) that on every compact set a continuous function is the limit of polynomials. But the preceding definition can be extended to a function f such that there exists x E H, x =F 0, such that f is measurable for the measure defined by A H (E;.,x, x) = IEAXl2 and square summable, i.e.:

Hence

Definition 6. If f is a complex valued function defined on IR, then f(A) is the operator defined by the formula


(3.110) f(A)x = fIR f(A)dE ... x, X E D(f(A» ,

where D(f( A» is the set of vectors x E H such that f is measurable with respect to the measure d ... (E ... x, x) = d;.IE;.xI2 and such that

(3.111) fIR If(AW dIE;.(xW < + 00 .

It is easily verified that if x and y E D(f(A)), then x + Y E D(f(A», and hence the linearity of the operator f(A) so defined. In order to make a constructive theory of functions of an operator we can restrict consideration to those functions which are measurable with respect to dt(Etx, x) for all x E H. Such functions are obtainable as simple limits of continuous functions. Denote by A 0 the set of such functions. Then we have

f+OO

Proposition 4. Let fE Ao and f(A) = -00 f(A)dE;.. Then for all x E D(f(A»

and all Y E H, we have:

(3.112) (f(A)x, y) = f~: f(A)d(E;.x, y) .

Proof We give an idea of the (easy) proof (for all the details see for example Akhiezer-Glazman [1], pp. 70-71). To start with, note that (3.112) has been proved earlier in the case where f is a continuous function (see Theorem 2). It is easy to establish (3.112) whenever f is a function belonging to A 0 and bounded, and obtained as a limit in L;x.)lR) (where (jx.y is the measure dt(Etx, y), x and y fixed) of a sequence of bounded continuous functions fn. Whenever f is an unbounded function in A 0, we put

{f(t) for If(t)1 ~ n

fn(t) = o for If(t)1 > n.

For all n, f. E A 0 and is bounded, thus is such that

(fn(A)x, y) = f ~: fn(A) d(E;.x, y) ;

Since fn - fin L;x.)IR), (jx,y = d(E.l.x, y), and we get (3.122). o Remark 3. 1. Note that if Xa,p is the characteristic function of the interval (a, P), a < p, then:

(3.113) Xap(A) = f~: Xap(A)dE.l. = r dE;. = Ep - Ea·

2. Iff E A 0' then f( A) is a projection if and only if f (A) takes only the values 0 and 1, that is if f is the characteristic function of a set C c IR which is measurable with respect to the measures {d;.(E ... x, x), x E H}.


3. It is easy to see that the set of functions of the operator A obtained by considering only the functions f E vi! 0 which are bounded, form a commutative algebra. lll In particular, E;. commutes with f(A). 0

More generally, the next theorem defines a functional calculus for functions which are not necessarily bounded.

Theo!:.em 11. Let A be a self-adjoint operator in the separable Hilbert space H. i) Iff is the complex conjugate function for f, then

D(j(A)) = D(f(A))

and for x, y E D(f(A)) = D(f(A)):

(3.114) (f(A)x, y) = (x,J(A)y) .

ii) Ifx E D(f(A)), y E D(g(A)), then

(3.115) f+OO

(f(A)x, g(A)y) = _ 00 f(A)g(A)d(E;.x, y) .

iii) For rx E 1[, X E D(f(A))

(3.116) (rxf)(A)x = rxf(A)x .

For x E D(f(A)) n D(g(A)), we have

(3.117) (f + g)(A)x = f(A)x + g(A)x .112

iv) Ifx E D(f(A)), then the conditionf(A)x E D(g(A)) is equivalent to the condition x E D(g.f(A)) (where g.f(A) = g(A)f(A)) and we have

(3.118) g(A).f(A)x = (g. f)(A)X .11 2

v) Iff E vi! 0 and D(f(A)) is dense in H, then the adjoint [f(A)]* of f(A) satisfies

(3.119) [f(A)]* = ](A) ;

f(A) is then normal (and self-adjoint whenever] = f). vi) Iff # 0 a.e. with respect to the measures {ax }xEH,113 then [f(A)r l exists and

(3.120) [f(A)r l = (} )<A). Proof i) it is immediate that D(f(A)) = D(j(A)) and that

(f(A)x, y) = f:: f(A)d(E;.x, y) = f:: f(A.)d(x, E;.Y)

= (f(A)y, x) = (x,](A)y).

III And also a commutative C* algebra. 112 Not that one only has f(A) + g(A) c (f + g)(A) and g(A)f(A) c (g. j)(A).

def 113 With the notation (Ix = (Ix.x'


ii) We noted earlier that E;,. commutes with A thus with polynomials in A and hence with f (A) for all f E vii 0 (by passing to the limit). Hence:

(f(A)x, g(A)y) = f:: f(Je)d(E;,.x, g(A)y) = f:: f(Je)d(x, E;,.g(A)y)

f+OO

= -00 f(A)d(E;,.g(A)y, x)

f +OO f+oo = -00 f(Je)d -00 g(J.L)d(E/lE;,.y, x)

= f::f(A)d f~oo g(J.L)d(y,E/lx) = f::f(Je)g(Je)d(E;"X,Y)

for all x E D(f(A)), y E D(g(A)). iii) is evident.

iv) Let x be such that f:: If(JeWdIE;,.xI 2 < +00. Since E;,..E/l = Einf(;,.,/l)'

the condition f:: Ig(JeWdIEJ(A)xI2 < +00 implies (from the fact that

EJ(A) = f(A)E;,.) that:

f:: Ig(JeWdlf(A)E;,.xI2 = f:: Ig(JeWd;,.IEJ(A)xI2( < +00)

= f:: Ig(JeW d;,. (f:: If(J.LW d/lIE/lE;,.xI2 )

Since the preceding calculation can be carried out in the reverse order, we see that under the hypothesis x E D(f(A)), the two conditions f(A)x E D(g(A)) and x E D(g.f(A)) are equivalent and we have:

(g(A)f(A)x, y) = f:: g(Je)d(EJ(A)x, y)

= f:: g(A)d(f~oof(J.L)d(E/lX'Y))

f+oo

= -00 g(Je)f(Je)d(E;,.x,y) = (g·f(A)x,y).

v) Iff is a function E vii 0 and bounded a.e. for the measures (J x: (J At) = dt( Etx, x), then D(f(A)) = H = D(j(A)) from i), and (f(A))* = l(A).


Further, from iv), it is easy to see that

f(A).(f(A))* = (f(A))* ·f(A)

thus f( A) is normal and self-adjoint if f takes real values. Let f be unbounded and D(f(A)) dense in H (otherwise the operator f(A) will have no adjoint. Let Y E D(f(A)*); then 'Ix E D(f(A)), 3y* E H,

(3.121) (f(A)x, y) = (x, y*) .

Denote by en = {t E IR; If(t)1 < n} and let

(3.122) fn(t) = {f(t), tEe. 0, t ¢ en .

Since J,,(A) is bounded, we have for all Z E H

(3.123) (J,,(A)z, y) = (z,].(A) y) .

Let Xen be the characteristic function of en and let E( en) be the projection corresponding to XeJ A). Then

def (3.124) the vector x = E(en)z belongs to D(f(A)) ;

in effect we have E;.x = E;.E(en)z = f~oo XeJt)dEtz, and

f:: If(AWdIE;.xI2 = f:: If(AW Xe.{A)dIE;.zI2

= f:: Ifn(AW dlE;.zl2 < + 00 .

From (3.121) and (3.123) we deduce:

{(Z,].(A)Y) = (E(en)z, y*)

(3.125) (z,].(A)y) = (z, E(en)y*) .

Since z is arbitrary in H we have

].(A)y = E(en)y*

(3.126) 1].(A)yl = IE(en)y*1 :( ly*1

It follows from (3.126), on letting n ~ 00 that

(3.127) f+oo -00 Ij(AWdIE;.yI2 :( IY*12.

We have thus proved that if y E D(f(A)*), then y E D(j(A)).


From point i), the converse is true and

(3.128) y* = j(A)y .

Whence (3.t19) and the fact that the unbounded operator f(A) is then normal. vi) (f(A))-1 exists if and only if f(A)x = 0 implies x = O.

f+OO

An equivalent condition is _ 00 If(.A.W dlE~xl2 = 0 ¢> x = 0 which holds if and

only if the set of.A. such that f(.A.) = 0 has measure zero for each O'x'

1 If this condition is satisfied then it is evident from iv) that the function 7 defines

(f(A))-I. 0

Application of Theorem 11. Spectral decomposition of a unitary operator. Let U be a unitary operator in H such that 1 tf (J p( U) (i.e. R(l - U) is dense in H). Then we know 114 that it is possible to associate with it a self-adjoint operator A in H through the formulae:

(3.129) {A = i[1 + U][I - Uri U = [A - iJ][A + ilr 1 ;

U is called the Cayley transform of A.114 From the spectral decomposition of A it is possible to deduce a spectral decomposition of U. Let E A be the spectral family for A; then if we put

(3.130) {.A. = - cotg(ej2), 0 < e < 2n E~ = Fe,

Fe is a spectral family for the unitary operator U satisfying:

(3.131)

and we can write

Fe is a projection operator

Fe·Fe' = Finf(e,e')

Fo = 0, F27t = 1 ,115

Fe+ 0 = Fe, 0 ~ e < 2n ,

(3.132) U = f: 7t eiedFe .

The spectral decomposition of U so obtained has properties analogous to those of the self-adjoint operator A which corresponds to it under the Cayley transformation. We note that the spectrum of U is simple if that of A is.

Remark 4. We have deduced the spectral decomposition of unitary operators from the spectral decomposition of self-adjoint operators by making use of the Cayley transformation.

114 See Chap. VI, §3. 115 Here we also have F 2 .- 0 = 1.


Naturally one can directly construct a spectral decomposition of unitary operators analogous to that of self-adjoint operators (see Riesz-Nagy [1]). For example, if $'

is the Fourier transformation, a unitary operator from LZ(/R) onto itself, we can note that:

(3.133) $'4 = 1.116

so that the formulae (3.129) do not make sense here (variants must be used instead) because -1 and + 1 are eigenvalues of $'.117 But having (3.133) we can introduce the operators:

(3.134) 1

Pz = 4. (l - $' + $'z - $'3)

1 P 3 = - (l + i$' - $' Z - i$' 3 )

4

which are pairwise orthogonal projections reducing the space L Z (/R), that is to say satisfying:

(3.135) { I = Po + P1 + Pz + P3

$' Pk = ik Pk , k = 0, 1, 2, 3 ;

$' has thus + 1, i, -1, -i for its eigenvalues, and its eigenvectors are the Hermite functions (see §2.7.4). The spectral decomposition of $' is thus of the form:

On defining Fe on [0, 2n] by:

0, if n

0~(}<2

P1 , if n 2~(}<n

(3.137) Fe = P1 + Pz , if

3n n~(}<T

P1 + Pz + P3 , if 3n 2 ~ () < 2n

I , if () = 2n

116 In effect. :Y' being unitary, we have :Y' 2 I = j with j (x) = 1(- x), because

.'F 1= (.#1) = (Y;-lj)~ VIE L2(lJ;l"). 117 Hence:Y' is not the Cayley transformation of any self-adjoint operator.


we have the representation

(3.138)

analogous to that obtained in (3.132). The same holds for each unitary transformation. 0

5. Operators which Commute with A and Functions of A

Definition 7. Let A be an unbounded operator in H, and B a bounded operator. We say that B commutes with A (or that A and B commute) if:

(3.139) {

for all x E D(A), we have

i) Bx E D(A)

ii) BAx = ABx ,

(which we write as: AB ::J BA).

We then have

Proposition 5. Let A be a self-adjoint operator in Hand B E !l'(H), and let {E;.}.( E IR be the spectral family associated with A; A and B commute if and only if Band E;. commute for all A E ~~.

Proof If A and B commute, then for all ( ¢ ~ we have:

B«(1 - A)x = «(I - A)Bx, x E D(A) .

Let R«() = «(I - A)-I; since R«()H = D(A), for all z E H we have:

R«()Bz = BR«()z, Vz E H .

By considering the integral representation of the resolvant,118 we have for all y, z E Hand ( E C \~:

(3.140) f +CO 1 f+co 1 -r=.- d(E,Bz, y) = -r=.- d(BE,z, y) . -coo, t -000, t

Whence we deduce:

(3.141)

and conversely, and Proposition 5 follows. o Remark 5. From our studies in §3.4 it follows that each bounded function of a self-adjoint operator A commutes with A. The converse is false in general: if B is an operator which commutes with a selfadjoint operator A, then B is not necessarily a function of A. 0

118 See (3.63).


Nevertheless, we have

Proposition 6. Let A be a self-adjoint operator with simple spectrum. If B E !f (H) commutes with A then B is a function of A.

Proof If the spectrum of A is simple there exists a generator y of A, that is to say, y E H such that the subspace spanned by the set {(Ep - Ea)y, iX < /3, iX, /3 E IR} is dense in H. Hence each vector x E H has a representation of the form 119

(3.142) f+OO

X = -00 ~(t)dEtY

where ~ E L;y(lR) with (1y = d(EtY, y) = d1EtY12. If B is an operator belonging to !f(H), then in particular

(3.143) By = f:: cp(t)dEty, cp E L;y(IR).

Assuming for the moment that

(3.144) cp is a function bounded a.e. with respect to (1 y •

Choose x E H. Then for each z E H, we have from (1.142)

f+OO f+oo

(Bx, z) = (x, B*z) = -00 ~(t)d(EtY, B*z) = -00 ~(t)d(BEtY, z)

f+OO f+oo

= -00 ~(t)d(EtBy, z) = -00 ~(t)d(By, Etz).

Since from (3.143)

f+OO ft

(By, Etz) = -00 cp(u)d"(E,,y, Etz) = -00 cp(u)d(E"y, z),

we have

f+OO

(Bx, z) = _ 00 ~(t)cp(t)d(EtY, z) ,

and

(3.145) Bx = f:: ~(t)cp(t)dEtY = f:: cp(t). ~(t)dEtY, "Ix E H .

Whence it follows that

(3.146) B = cp(A) .

The theorem is thus proved as soon as we have demonstrated (3.144).

Proof of(3.144). Suppose that cp is unbounded; then there exists a sequence of sets of measure non-zero for (1 y such that I cp( t) I > n, tEen'

119 See for example Akhiezer-Glazman [1], p. 279, Theorem 2.


def If E(en) = Xe)A) denotes the projection defined by the characteristic function Xen

for en, consider the elements ~n = E(en)y; ~n '# 0 because

l~nl2 = (E(en)y, y) = mes(1) {en}) '# O.

Now, see Remark 6 below, if B commutes with A, then B commutes with every function of A, thus B commutes with E(en) = Xe)A). Hence:

and

whence:

BE(en) = E(en)B,

IB~nI2 = IBE(en)yI2 = IE(en)ByI2 = (By, E(en)By)

f +OO f+oo = -00 cp(t)d(ErY, E(en)By) = _ 00 X.e)t)cp(t)d(ErY, By)

= f~: Xn(t)lcp(tWdIEryI2 > n2 f~: Xe)t)dIEtyIZ,

IB~nI2 > n21~n12 ,

which implies that the operator B is unbounded; whence (3.144) holds and Proposition 6 is proved. 0

Remark 6. i) We observe that a necessary condition for an operator B E !£,(H) to be a function of the self-adjoint operator A is:

(3.147) B commutes with every bounded operator T which commutes with A.

In effect, if B = f(A) with f a function on a(A), and if T commutes with A, then

(3.148) E;. T = TE;., V A E IR ,

where {E;.} 2 ElKl is the spectral family associated with A. It follows that for all x,y E H,

f+OO f+oo

(f(A)Tx, y) = -00 f(A)d(E;.Tx, y) = -00 f(A)d(TE"x, y)

f+OO = _oof(A)d(E;.x,T*y) = (f(A)x,T*y) = (Tf(A)x,y) ,

whence

(3.149) f(A). T = T·f(A).

In fact, condition (3.147) is also sufficient for B to be a function of A, but the proof of this fact is more complicated than that for the necessity of the condition. ii) In order to be able to state the most general result (Theorem 12 below) we use the notation:

(3.150) {(A)' the set of B E !£,(H) which commute with A (i.e. which satisfy (3.139» .


Then we have:

(3.151 ) (f (A))' :;) (A)'

which contains (3.149) as a particular case. More generally, we shall simply state the following result (due to Neumann, Riesz, Mimura) the (highly technical) proof of which can be found for example in Yosida [1], p- 340.

Theorem 12. Let A be a self-adjoint operator in the separable Hilbert space H. Let T be a closed operator with domain D( T) dense in H. Then a necessary and sufficient' condition that T = f(A) iS120

(3.152) (T)' :;) (A)' .

6. Fractional Powers of a Strictly Positive Self-Adjoint Operator121

Let A be a self-adjoint operator in the separable Hilbert space H with domain D(A) dense in H satisfying (3.73) (that is to say):

(3.73) { there exists a constant IX > 0 such that (Ax, x) ~ IXjxj2 for all x E D(A)) .

We then know (Proposition 3) that the spectral family {E.d..! E ~ associated with A satisfies (3.74) (i.e. E;,. = 0 for A < IX). Thus, taking 0 < Ao < IX, P E [0, 1], the function A 1--+ AP (defined on [A. ° , + 00 [ with values in JR) allows us to define AP by:

(3.153) ;"0 {

AP = f+oo APdE;,.

D(AP) = {x E H; f:ooo A2P djE;,.xj2 < +00 }.

For all x E D(AP) and y E H, we have:

{ (APx, y) = f:ooo

APd(E;,.x, y)

(3.154) jAPx j2 = f+oo A2P djE;,.xj2 .

;"0

Denote:

for p = 0, D(AP) = H, AP = I

for p = 1, D(AP) = D(A), AP = A .

120 With f not necessarily bounded on a(A). 121 That is to say, positive and possessing a bounded inverse.


We then have the following theorem:

Theorem 13. Let A be a self-adjoint operator with domain D(A) dense in H, satisfying (3.73). For 0 ~ P ~ 1, we have, with the notations of(3.153): i) D(A) c D(AP) (thus D(AP) is dense in H);

ii) for all x E D(AP),

(3.155)

Furthermore, (AP)-l = A -P E .!£(H); A -P and AP are self-adjoint; iii) a) D(AP) given the norm defined by:

(3.156) Ixl; = Ixl2 + IAP x I2, X E D(AP) ,

is a Hilbert space;

b) if 0 ~ PI < P2 ~ 1,

(3.157) D(AP2) c+ D(AP') (C+ continuous injection)

and D(AP2) is dense in D(AP'); iv) for all x E D(A), we have:

(3.158)

Proof i) Let x E D(A) then for 1 ~ 111 < 112 we have:

Since x E D(A), lim fll2 ,FdIE"xI 2 = 0, which implies 1'1 --. 00 JJl

lim fll2 A. 2P dIE"xI 2 = 0, whence x E D(AP). P,1 - 00 1'1

Since D(A) is dense in H, so also is D(AP). ii) Note that for x E D(AP)

(AP x , x) = f+oo A.Pd(E"x, x) ~ A.g f+oo dlE"xl 2 = A.glxl 2

"0 0

which holds for all A.o E JO, a [, whence (3.155). From (3.155), (AP)-I exists and belongs to '!£(H).

If we introduce A-P = f+oo A.-PdEA, we see that

"0

D(A-P) = {XEH; f:o oo A.- 2P dIE"xI 2 < +oo} = H

122 Or directly, because the function A f-+ A -2p is bounded on Po, + 00[.


Furthermore, A -P E 2(H) and (A -P)* A -P because l23

(A - P x, y) = f + 00 A - P d ( E;. x, y) = f + 00 A - P d ( E;. y, x) = (A - P y, x) ;'0 ;'0

= (x, A -P y ) .

Finally, (A -P AP x , y) = (x, y),

since for x E D(AP), y E H

but

Therefore we have

and hence A -P = (AP)-l ;124 AP, which is thus the inverse of A -P (which is selfadjoint) is itself self-adjoint, and thus in particular closed. iii) a) First of all the norm I Ip corresponds to the scalar product

(x, y)p = (x, y) + (AP x, APy ) ,

and thus D(AP) is a pre-Hilbert vector space. If Xn is a Cauchy sequence for lip, Xn and AP xn are Cauchy sequences in H; H being complete, Xn -4 x and APxn -4 xP in H; A being closed, x P = A P x. Further, Xn -+ x in D(AP) which is thus a Hilbert space. b) If 0 ~ P1 ~ P2 ~ 1, we have for x E D(AP2):

f+OO

A2Pl dlE;.xl2 ;'0

hence for P2 ~ P1

X E D(AP2) .

123 Or alternatively, see Theorem 11. 124 See also Theorem 11 (using f(A) = AP in the proof).


To show that D(AP2) is dense in D(API) for Pz > PI' it suffices to show that for all pE ]0,1[, D(A) is dense in D(AP). If R(z) = (z1 - A)-I is the resolvant for A, then for x E D(AP) put:

(3.159) ( 1 )-1 Xn = 1 + ~A x = n(n1 + A)-IX = - nR(-n)x.

Therefore Xn E D(A) and since R( -n) is self-adjoint and commutes with E;. (as a function of A) we have (for all y E H):

(AP xn , y) = f+oo - A,Pd(E;.nR( -n)x, y) = f+oo A,Pd(E;.x, - nR( -n)y) ~ ~

= (AP x , - nR( -n)y) = (-nR( -n)AP x , y) ;

thus

(3.160)

It is easy to verify that from (3.159)-(3.160)

xn -+ x and A P Xn -+ A P x in H as n -+ + 00 ,

and thus that Xn(E D(A)) -+ X(E D(AP)) in D(AP). 1 1

iv) From the Holder inequality with exponents p = -, q --, we have: P 1 - P

~ (f: oo A,zdIE;.xlz Y(f:o oo

dlE;.xlz Y-p which is (3.158). o An application: spaces intermediate between Hilbert spaces. We consider the following situation which frequently appears in applications. Let X and Y be two complex separable Hilbert spaces; we assume that:

(3.161) {X c; Y.125

X is dense in Y.

Denote by ( , h (resp. ( , }y ), I Ix (resp.1 I y) the scalar product and norm in X (resp. in Y). Put

(3.162) a(u,v)=(u,vh for U,VEX.

Let A be the unbounded operator l26 in Y with domain:

(3.163) D(A) = {UEX; vf---+a(u, v) is continuous on X for the topology on Y}

(D(A) is dense in X and in Y). We know lZ6 that A is a positive self-adjoint operator with bounded inverse

125 I.e. X is included in Y under continuous injection. 126 See Chap. VI.


(0 ¢ O"(A)), satisfying:

(3.164) { (Au, u)y = a(u, u) = lull ~ alulL a > 0 (from (3.l61)-i))

for all u E D(A) .

Let {E/l}/i E Ihl be the spectral decomposition of A in Y (in fact O"(A) c [Po, + 00 [

with 0 < Po). For u E D(A) we have:

thus:

(3.165)

From the denseness of D(A) in D(A 1/2) and X, we see that

(3.166) D(AI/2) = X.

def Put A = A 1/2; A is also a self-adjoint operator with domain D(A) = X satisfying:

(3.167) (Au, u}y ~ a1/21ul~ for all u E D(A) = X .

Let {F.,} be the spectral family associated with A; we have

(3.168)

We now make

Definition 8. Let X and Y be two separable Hilbert spaces satisfying (3.161). For () E [0, 1], we put

(3.169)

where A = A 1/2 is the operator unbounded in Y, with domain X, defined above. Given the scalar product:

(3.170)

[X, Y]o is a Hilbert space (see Theorem 13).

Then

(3.171) {[X, Y]o = D(A) = X

[X, Y]I = D(I) = Y;

[X, Y]o is a space intermediate between X and Y. It is called l27 the holomorphic interpolation of order () between X and Y.

These spaces have the following interpolation property: every linear operator continuous from X into itself and from Y into itself is also continuous from [X, Y]o into itself, for 0 < () < 1 (see §4 below).

127 See Lions-Magenes [1].


Example 4. Let X = Hm(Q), Y = L2(Q); then one can define the space HS(Q), for s E [0, m] by:

with (1 - O)m = s (see Lions-Magenes [l],·Chap. 1.)

From Theorem 13 we deduce

Proposition 7. With the hypotheses of Definition 8, we have: i) for 0 E [0, 1], there exists a constant C( 0) > 0 such that for all u E X

(3.172) lul[x,y]. ~ C(O)lull-o.lult;

ii) for 0 ~ 01 < O2 ~ 1

(3.173) {[X, Y]O, G [X, Y]02

[X, Y]8, is dense in [X, Y]02

o

Proof Point ii) being evident from Theorem 13, we prove point i). From (3.170) and (3.158) we have

lul~ = lul~ + 1...1 1 - 8ul~ ~ lul~ + IAul~(1-8) luW

~ luW [lul~(I-O) + IAuW I - 8)].

From (3.165) we thus have:

lul~ ~ luW[luWl -8) + lul~(1-8)] ,

and from (3.161) there exists a constant C = I/O( with

IUly ~ Clulx for all u EX, so that

whence (3.172). o Remark 7. The notion of spaces intermediate between Hilbert spaces (or indeed Banach spaces) is used for example in the study of evolution problems (see, for example, Chap. XVIII). i) In effect one encounters the following situation: let t H u(t) be a function such that

(3.174) { uEL2(IR;X)

du 2 •. X, Y satisfying (3.161) ; dt E L (IR, Y) ,

then it can be shown that u is the class of continuous functions on IR with values in [X, yr/2 .

ii) Another situation, relative to 2nd order evolution equations, is the following: a vector function t H u(t) is such that:


(3.175) { U E L2(/R; X)

d 2 u 2. X, Y satisfying (3.161) . dt 2 E L (/R, Y) ,

Then one can deduce by interpolation that

(3.176) du 2 . Cit E L (/R, [X, Y]1/2) .

Hence, from i), u will be the class of a continuous function on /R with values in

[X, [X, Y]1/2]1/2 = [X, D(A 1/2)]1/2 = [X, Y]1/4,

and, again from i), du/dt will be the class of a continuous function on /R with values In

[[X, Y]1/2' Y]1/2 = [X, Y]3/4·

The situations considered here form a special case of a more general interpolation theorem treated in §4 below. 0

§4. Hilbert Sum and Hilbert Integral Associated with the Spectral Decomposition of a Self-Adjoint Operator A in a Separable Hilbert Space H*

Let H be a complex separable Hilbert space (with scalar product ( , ), and norm I I) and let A be a self-adjoint operator with domain D(A) dense in H. We now propose to extend to the general case the results on the decomposition of H into a Hilbert sum obtained in §2 (see §2 and the introduction to §3, formula (3.1» in the case where the self-adjoint operator was compact or had a compact resolvant.

1. Canonical Representation Associated with a Self-Adjoint Operator Whose Spectrum is Simple

Let A be a self-adjoint operator whose spectrum is simple. lf { E A} ). E IR is the spectral family associated with A we recall that there exists at least one vector Yo E H such that the vector subspace generated by the set

{(Ep - Ea)YO.; IX, fJ E /R, IX < fJ}

is dense in H. Such a vector Yo is called a generator or cyclic vector. Put

(4.1) { for all A. E /R

p(A.) = (EAyo, Yo) = IE AYol 2 •

The function p: A. ~ p(A.) defined by (4.1) is an increasing monotone function in the wide sense, right continuous and bounded, (0 ~ PO.) ~ IYoI2, V A. E /R) whose derivative in the distribution sense, denoted by dp, is a measure.

§4. Hilbert Sum and Hilbert Integral 155

Denote by L;(IR) the Hilbert space of (classes of) complex valued functions square summable on IR for the measure dp. If fE L;(IR), we denote by

Iflp = (f ~ If(AW dp(,l.) yl2 the norm of j, and by

(f, g)p = f ~ f(,l.)g(,l.) dp(A)

the scalar product of j, 9 E L;(IR). Recall that the measure of an interval ]a, b] of IR, p(]a, b]), is then

p(]a, b]) = f ~ Xja, bjdp = p(b) - p(a) ,128

and that f belongs to the class of 9 if f(x) = g(x) p - almost everywhere (which we denote by p-a.e.). With each fE L;(IR), we associate the vector JE H through

~ def def f+ 00

(4.2) f= ilJI-lf= _oof(,l.)dE~Yo

Then we have

Theorem 1 (Theorem of diagonalisation of the operator A).129 The operator ilJI defined by (4.2) is an isometry from the Hilbert space H to the Hilbert space L;(IR) such that

(4.3)

is the operator of multiplication by ,l. in L;(IR) with domain

(4.4)

Proof Since E;.EI' = Einf (;',l'j' we have 'rj f E L;(IR):

(E;.Yo, J) = f~: f(fJ.) dl'(E;.Yo, EI'Yo) = f~: f(fJ.) dl'(EI'E;.Yo, Yo)

= f~oo f(fJ.) d(EI'Yo, Yo) = f~oo f(fJ.) dp(fJ.) ,

so that for all f and 9 E L;(IR):

(j,g) = f~:f(,l.)d(E;'YO,g) = f~:f(,l.)9(A)dP = (j,g)p.

128 Note that p([a, b)) = p(b) - pta - 0). 129 Recall that one of the hypotheses is that the spectrum of the self-adjoint operator A is simple.


H:n~e 0!t- 1 defined by (4.2) maps L~(lI~) linearly and isometrically onto {J; f defined by (4.2)} = Hie H. In particular, HI is a closed subspace in H containing elements of the form

J: dE;.Yo = (Ep - Ea)yo, ex, {3 E IR, ex < {3 yo

The spectrum of A being simple, we can deduce that HI = HI = H. We now establish the second part of Theorem 1. First we note that V f E L~(IR):

(4.5) EJ = E;. f ~ f(J-l)dEllyo = f~: f(J-l)dll(E;.E/lYo) = f~ (x,!(J-l)dE/lYo ,

so that for all f and g E L~(IR):

(4.6) (EJ, g) = f~oo f(J-l)g(J-l)dp .

~ence the condition f ~ ..1. 2 dlEJI2 < + 00, which is equivalent to the condition

fE D(A), is equivalent to:

Furthermore,

AO!t- 1 f= Aj= f~: A.dE;.j,jED(A),

so that from (4.5H4.6),

(BJ,g)p = (O!tAO!t- 1 J,g)p = (AO!t- 1 f,0!t- 1 g) = (Aj,g)

= f~A.d(EJ,g)= f~:A.f(A.)g(A.)dP' VjED(A), gEH.

Moreover, we have (with the same conditions on f and g)

f+oo (Bf, g)p = -00 (Bf)(A.)g(A.) dp

whence it follows that we must have

(Bf)(A.) = A.f(A.), p-a.e.,

whence Theorem 1. 0

Example 1. Let H = L2(1R), A the operator of multiplication by x (which d

corresponds to the observable "position" in quantum mechanics 131), B = i dx

130 Obtained by taking the image under uti - 1 of the characteristic function X,p of the interval ]tx, fJl 131 See Chap. lA. §6.2.


(which corresponds to the negative of the "momentum" observable 131 ). We have previously noted 132 that:

(4.7) {A = :F- 1 B:F B=:FA:F- 1

where OU = :F is the Fourier transform. o Example 2. Let A be a positive self-adjoint operator in H such that A - 1 is compact with simple spectrum. Then we know that

n ~ 1

v,. = {C tPn} where tPn is an eigenvector of A for the eigenvalue )'n (0 < Al < A2 < ... < An < ... ) such that ItPnl = 1. A generator vector Yo (also called cyclic )133 is

(4.8) 6 00 1

IYol2 = 2 L 2 = 1 1t n~ 1 n

(in effect, EA being here the projection on GA = EEl Vn: we have: n

An ~ A.

(EA - EA -I)YO = j6tPn,n ~ 1 withtheconvention EAO = 0, n n 1t n

00

so that if Y = L an tPn E H then n~l

We then have

(4.9) 6 00 1

dp(A) = d(EAyo, Yo) = 2 L 2<>{}- - An) 1t n~l n

(dp( A) is the measure defined by the "masses 62 ~ concentrated at the points An") . 1t n

132 See (3.92). 133 See the start of §4.1.


An elementJE L~(IJ~) will here be any function Jdefined at the points Pn} such that

An elementJ E L~(lR) is thus the class of a sequence offunctions square summable

for the "weights" ~, i.e. n

f I.t,;r < +00, n~ 1 n

and

_ (~ 00 I.t,; 12 )1/2 IJlp - 2 L 2

1t .~1 n (4.10)

In this case we will have:

(4.11)

Conversely,

00 _

for j = L .t,;lP. E H (~L lin 12 < + (0) n~l

(4.12) 1t -

O/IJ = (O/If)nEN*; (O/If). = .j6 n.t,; .

and B = 0/1 A 0/1-1 is defined by:

(4.13) { JE L~(IR) = (.t,;)nEN* (Bf). = A • .t,; for all n = 1, ....

o

In the following section we shall generalise the result of Theorem 1 to the case of an arbitrary self-adjoint operator A.

§4. Hilbert Sum and Hilbert Integral

2. Hilbert Sum Associated with the Spectral Decomposition of a Self-Adjoint Operator A in a Separable (and Complex) Hilbert Space H

159

Let A be a self-adjoint operator in the Hilbert space H, and {E.dAEIR its spectral family.

Definition 1. Let x E H. The closed subspace of H generated by .the set {E;.x, A E IR}, and denoted by H x , is called the cyclic subspace of H generated by x.

We note that if y is a vector in H, orthogonal to Hx, then Hx and Hy are orthogonal because if A, p. E IR then

(E;.x, Elly) = (Einf (;',Il)x, y) = O.

We now show that it is possible to obtain a decomposition of H into a Hilbert sum of cyclic subspaces. For this, we note that from the separability of H, there exists a sequence S = {vn}n dense in H. We put

(4.14) VI WI = ~' HI = HW"

Then suppose that we have constructed p pairwise orthogonal cyclic subspaces HI' ... ,H p of H. Let vnp + I be the first of the vectors k = 1, 2, ... such that

p

(4.15) vnp + I ¢ Gp = EEl Hk • k~1

Then in the vector subspace generated by G p and vnp + I' we can find a vector w p + I

satisfying

(4.16) { lwp + 11 = 1

w p + 1 is orthogonal to G p •

Now putting

(4.17)

we have:

(4.18) Hp+ 1 is orthogonal to Gp .

Hence, by recurrence, we obtain a family {H p} pEN" of closed pairwise orthogonal subspaces of H. By construction, the vector subspace generated by the union of the H p contains the set Sand:

00

(4.19) H = EEl Hn· n~l

We then have the

Proposition 1. The spaces Hn defined by (4.14)-(4.17) possess the following properties:


i) '<Ix E Hn, E;..x E Hnfor any A E ~; ii) if Pn is the projection onto Hn,

(4.20)

Proof. i) Let X;. = E;.. Wn where Wn is the "generator" of Hn satisfying (4.16)-(4.17). Then

E,..x;. = E,..E;. Wn = Einf (,...;.) Wn E Hn ,

whence i), by the denseness of the vector subspace generated by the X;.. in Hn and from the continuity of Ep., we have JJ. E ~.

ii) If Pn is the orthogonal projection of H onto Hn and X E H we have:

00

PnE;.x = Pn E;. L Ppx; p=l

from the continuity of E;.. in H: 00

PnE;..x = Pn L E;.Ppx, p=l

and from i) E;..Ppx E Hp, consequently

PnE;.x = PnE;.Pnx = E;. Pn X, '<Ix E H whence ii). 0

Example 3. Here we make use again of Example 2, Sect. 4.1. With the same notations, we can take: '

(4.21)

Definition 2. With the notations (4.14)-(4.19),/or n E N* put

(4.22) Pn(A) = (E;,wn' wn),

o

and denote by L~J~) the Hilbert space of (classes of)functions square summable with respect to the measure dPn. Further, let X;. be the function defined by

(4.23) { I if t::;:;A 2

X;.(t) = 0 if t > A; X;. E LpJ~).

We then have

Proposition 2. Under the hypotheses of Definition 2,/or all n E N *, there exists a unique unitary operator IlIinfrom Hn onto L~J~) such that:

(4.24) IlIin(E;.. wn) = X;., '<I A E ~ .


This mapping preserves scalar products since:

(x)., X/L)Pn = f:: x).(t)X/L(t)d,(E,wn, wn) = foo d,(E,wn, wn)

= (E). W n , wn ) = (x)., x/L) for A < 11 .

Since the vector space generated by the {x).} is dense in Hn and since the vector space generated by the X). is dense in L; (IR), we can extend diin to a unitary operator OlIn from Hn onto L;JIR). Its uniqueness is immediate from (4.24). 0

Put

00

(4.25) H = EB L; (IR) . n= 1 n

We have the following result (the analogue of that in Theorem 1 which treated the particular case of a self-adjoint operator A with simple spectrum).

Theorem 2. Under the hypotheses of Proposition 2: i) the operator OIl from H onto H defined by:

{ f =. {In) _E H, OIIf = 1 =*L!n} E H (4.26) ~ wIth In - OIInln, \In EN,

is a unitary operator from H onto H; ii) the operator

(4.27)

is the multiplication by A operator in H; i.e.

D(B) = {l = Jl i.. E H; Ai.. E L;JIR), n E N*

(4.28) with ntl Ai.. convergent in H}

Proof i) Firstly, OIl is a bijection; moreover,

00 00 00

I OIIfl~ = L 1i..1; = I I OIInln I; = I lin 12 = Ifl2 n:=l n n=l n n=l

whence point i). 00 00

ii) Step 1. Let x = I Xn = I Pnx E H. We show that

(4.29)

n=1 n=1

{

a) x E D(A) <:> {xn E D(A) for all n E N*

with L IAxnl 2 n

b) APn ~ PnA for all n E N * .

< oo} ,


In effect, if x E D( A) for ct, f3 E IR we have: r A2d(E;.xn' xn) = r A2d(E;,xn' x) (and by Cauchy-Schwarz)

~ ( r A2d(E;,xn' xn) yl2 ( r A2d(E;.x, x) yI2 ,

whence we deduce: r ).2d(E"xn, xn) ~ r ).2d(E"x, x) ;

as f3 ~ + 00 and ct ~ - 00 the integral on the right hand side exists, with value

Conversely, if Xn E D (A) for all n and II AXn I; converges, then n f:: A2d(E;.x, x) ( = ~ IAxnl; )

exists and x E D(A) whence point a) in (4.29). 0

Hence Xn = Pnx E D(A) for all x E D(A) . If Y E H we have, taking into account that E;.Pn = PnE;. (see (4.20» ,

(Axn, y) = t Ad(E;.Pnx, y) = t Ad(PnE;.x, y) = fIR Ad(E;.x, Pny)

= (Ax, p.y) = (P.Ax, y),

whence point b) in (4.29). 0

Step 2. The following property holds:

{ x.ED(A) for all nEN* with LIAxnl2 < 00

x E D(A) is equivalent to • 00

Ax = L Ax. in H . • =1

(4.30)

In effect, if x E D(A), then Xn E D(A) and we have:

N N N LAx. = L AP.x = L PnAx = llN(Ax) 1 1 1

N N

where llN = L p. is 'the orthogonal projection onto GN = EB Hn· 1 1

00

Since N ......... GN is an increasing sequence, U GN = H, N= 1

we thus have, with respect to the norm on H:

llN(Ax) ~ Ax for N ~ 00 .


Conversely, let 00 00

x = L Xn E H with Xn E Hn n D(A) and L AXn convergent in H to y . n=l n=l

Then

and

N

(N = LXnED(A)'(N --+ x in H 1

N

A(N = L AXn --+ y in H. 1

Since A is self-adjoint and therefore closed, we conclude that

x E D(A) and y = Ax whence (4.30) .

Step 3. Now let

(4.31)

Then

(4.32)

and we have

(4.33)

X E Hn with OUnx = f.

X E D(A) is equivalent to AfE L;JIR)

OUn(Ax) = Af·

We prove (4.32) and (4.33). Assuming (4.31) we can write:

whence

L A2 d(E).x, x) = f A2 If(AW dPn(A) ,

which establishes (4.32).

163

o

Then let x E Hn n D(A) with OUnx = fand y E Hn with OUny = g; on one hand we have:

on the other:

(Ax, y) = r Ad(E).x, y) = r Ad). f). f(t)g(t)dPn(t) J~ J~ -00

= L Af(A)g(A) dPnO·) = (Af, g)p" = (Af, OUnY)p"

whence (4.33).

Step 4. (4.28) follows immediately from (4.30), (4.32), (4.33), whence the theorem. 0

Corollary 1. OU is an isomorphism from D(A) given the graph norm, onto D(B) (where B is defined by (4.27) and (4.28)) given the graph norm

1111(B) = 1111 + IA111 = f lin I;" + f lAin I;" . n=l 1


The proof is immediate.

Corollary 2. Suppose the operator A further satisfies:

{there exists a constant a > 0 such that

(4.34) (Ax, x) ~ alxl2 for all x E D(A) .

Then for all () E [0, 1], the operator

(4.35)

is the operator of multiplication by )..0 in H:

(4.36)

D(Bo) = {i = n~l i. Eli; )..01. E L;jlR) ,

'<In E N *, ~ )..01. convergent in H} Boi = f Boi. = f ),0 i. convergent in Ii .

n=l n=l

o

Furthermore, UU is an isomorphism from D(A 0) given the norm I 10 defined by §3,(3.l56), onto D(Bo) given the norm II 110 defined by:

(4.37) lIill~ = n~l 1,:00 ).. 2°1i.()")1 2dPn()..) (0 < )..0 < a).

Proof To establish (4.36) it suffices to repeat word for word the proof of Theorem 2, replacing A by A 0, ).. by )..0 in (4.30)-(4.33). To establish (4.37) we note that on the one hand:

Ixl~ = Ixl2 + IA oxl 2 ~ IA oxl 2 = Ilill~ and on the other, in virtue of (3.155), that

Ixl~ ~ C(()llill~ .

Remark 1. The measures dPn, n E N *, are bounded Radon measures. 134

o

In effect, if ffJ: ).. f-+ ffJ()..) is a continuous function with compact support, we have:

IdPn(ffJ)1 = l<dPn,ffJ)1 = If:: ffJ()..)dPn()..) I

= If:: ffJ()")d(EAWn,Wn)1 ~ IlffJll00·lwnl2

whence II ffJ II 00 = suplffJ()..)I·

Then

(4.38) sup I < dp., ffJ) I 11'1'11, ,,;; 1

II dp. II ~ 1 for all n E N * .

134 See the Appendix "Distributions" in Vol. 2.

o


Furthermore, let p. be the measure defined by:

(4.39) d - ~ dPn II d II 1 p. - nf-l 2nlldPnll' p.::(.

This, in the same way as dpn, is a positive bounded Radon measure; moreover, every dp.-null set is also a dpn-null set for each n. It then follows from the Lebesgue-Nikodym theorem, (see for example, Bourbaki [1], Integration, Dieudonne [1], Schwartz [1]) that we can prove:

Proposition 3. With the hypotheses of Theorem 2, there exists a positive bounded Radon measure dp. (given by (4.39» and for every n, a dp.-integrablefunction <Pn' such that

(4.40)

Denote by

(4.41)

{ .~) <Pn(t)! 0 dp.-almost everywhere n) dPn - <Pn dp. .

Fn = {t E IR; <Pn(t) > O} ,

and let L;(Fn) be the space of (classes of) functions square summable with respect to dp. on Fn , and put

(4.42) 00

iI = EB L;(Fn)' n=1

Then let C be the multiplication by A operator in fj defined (like H) by:

(4.43) _ 00 _

Cf = L AI.. n=1

We then have

Proposition 4. With the hypotheses of Theorem 2, there exists a unitary operator j/ from fj onto iI (iI is defined by (4.42)) such that

(4.44) C = j/ Hj/ -1 (C defined by (4.43)) ;

j/ is an isometry from D(H) given the graph norm onto D(C) given the graph norm.

Proof The operator j/n defined by

(4.45)

isometrically maps L;JIR) onto L;(Fn) and preserves multiplication by A. j/ is then defined in an analogous manner to 0/1. 0

Proposition 5. If A satisfies (4.34), the measure dp. has (as has each of the measures dPn' n E N*) its support contained in ],10' +00[,0 < ,10 < IX.


For e E [0, 1J, CO = "YBO"Y- 1 is the multiplication by 20 operator in iI . The proof of Proposition 5 is immediate.

3. Hilbert Integral. Diagonalisation Theorem of J. von Neumann and J. Dixmier

o

We now, with the help of the notion of a Hilbert integral given in a canonical form due to J. Dixmier, move on to the J. von Neumann diagonalisation theorem (Theorem 4). We content ourselves here with only an outline of the construction. For the technical details we refer primarily to Huet [lJ, then to Reed-Simon [lJ, Dixmier [lJ, Gelfand-Vilenkin [lJ, Vilenkin [1].

Definition 3. i) A mapping x 1--+ ~(x) defined on IR, where for all x, ~(x) is a complex Hilbert space with scalar product (resp. norm) denoted by ( , )x, (resp. I Ix), is called a field of Hilbert spaces. ii) A vector field is then a mapping x -> cp(x) defined on IR such that for all x, cp(x) E ~(x).

The set of all vector fields forms a complex vector space. iii) Let v be a positive Radon measure on IR. A v-measurable field of Hilbert spaces is afield of Hilbert spaces such that there exists a vector space 9Jl of vector fields which depend" v-measurably on x ", that is to say, satisfying:

i) for all cp E 9Jl, x 1--+ Icp(x)lx is v-measurable; ii) if the vector field cp is such that for all t/I E 9Jl,

(4.46) x 1--+ (t/I(x), cp(x))x is v-measurable, then cp E 9Jl; iii) there exists a sequence {cp"}" E I'J.' CPn E 9Jl for all n such that for

all x E IR, the sequence {CPn( x)} is total in the space ~ (x) .

Definition 4. A vector field f on IR is said to be square v-integrable if it is vmeasurable and if

(4.47) I If(x)l;dv(x) < +00 .

It is easy to verify that the set of square integrable vector fields is a complex vector space. On identifying two vector fields which are equal v-a.e., we obtain the space of (classes of) square v-integrable vector fields, denoted ~, which is a Hausdorff preHilbert space whenever it is given the scalar product:

(4.48) (f, g).Jf{' = fIR (f(x), g(x))xdv(x) .

We then prove

Theorem 3. ~ given the scalar product (4.48) is a Hilbert space.

Proof(rapid) of Theorem 3. (see Dixmier [1]). Let {in} be a Cauchy sequence in ~.


By extracting a subsequence, we can assume that 00

(4.49) L 1.[.+1 - In I.!!" < +00, n=1

whence we deduce that except perhaps for x E Z such that v(Z) = 0, we have: 00

(4.50) L IIn+I(x) - In(x)lx < +00, x rj: Z. n=1

Then for x rj: Z the series: 00

J;(x) + L In+I(X) - In(x) n=1

con verges in Jt" ( x ). N ow let f be the vector field defined by:

{J;(X) + t (In+I(X) -In(x)) ,

f(x) = n-I

o if XEZ;

xrj:Z

(4.51)

f is v-measurable being the limit v-a.e. of a sequence of v-measurable fields In. Furthermore,

(fIR If(xWdv(x)YI2 ~ IJ;I.!!" + J1IIn+1 -Inl.!!" < +00

so that f E Jt". Finally: lim If - fN I.!!" = 0 since

+00

If - fNI ~ L IIn+1 - In I.!!" . n=N

o

Definition 5. The space Jt" is called the Hilbert integral of Jt"(x) and denoted by

(4.52) Jt" = fel Jt"(x)dv(x).

Remark 2. If H is a given Hilbert space, let Jt"(x) = H, \Ix E IR and let v be the Lebesgue measure. Then the Hilbert integral of Jt"(x) is the space L2(1R; H) which we have encountered earlier (see Remark 2 in §3) and which we will often utilise in evolution problems (see especially Chaps. XV, XVI, XVII and XVIII). 0

We now construct a Hilbert integral Jt" = fel Jt"(t)dJl(t) associated with the

measure Jl defined by (4.39). The set Fn and the functions C{)n having been defined by (4.40)-(4.41), we introduce the sets:

(4.53) Ko = {t; C{)n(t) = 0, \In E N *} (satisfying Jl(Ko) = 0) ,

(4.54) Kn = {t E IR; n(t) = n}, for n ~ 1,


where n(t) is the number of values of k = 1,2, ... for which t E Fk • The function t H net) and the sets Kn are Il-measurable. Now let X be a complex separable Hilbert space and (~I' ... , ~n' ... ) an orthonormal basis for X. Let Xn = gl' ... '~n} be the vector subspace of dimension n, generated by (~I' ... '~n)' We denote by ( , h (resp. I Ix) the scalar product (resp. the norm) in X. Now consider the field of Hilbert spaces on IR defined in the following manner:

(4.55) {t H £'(t) = Xn(t) for t ¢ Ko £'(t) = {O} for t E Ko .

We show that this field is Il-measurable, by proving that the vector space of vector fields, 9)1, constituted by:

(4.56) 9)1 = {t H q>(t) E £'(t) , q> is Il-measurable from IR -+ X}

satisfies the conditions (4.46). In effect, (4.46) i) follows from 1q>(t)lt = 1q>(t)lx and from (4.56). Let q>: t -+ q>(t); q>(t) E £'(t) be a vector field on IR such that we have:

(4.57) {t H (q>(t), I/J(t))t = (q>(t), I/J(t)h is Il-measurable for all I/J E 9)1 .

Since X is separable, it suffices to show that, for all U E X, the function t H (u, q>(t)h is Il-measurable.

00

Put u = L Ui ~i E X, and let I/J n be the function defined by i= 1

(4.58) { t Ui~i if t E Kn , I/Ju(t) = i=1

o if t E Ko .

Then I/J u E 9)1, and satisfies:

(4.59) (U, q>(t)h = (I/Ju(t), q>(t))t .

n E N*

It then follows from (4.57)-(4.59) that q> is Il-measurable, whence (4.46) ii). Next we show that (4.46) iii) is satisfied. To do this we define the sequence of vector fields {cn (t)} by:

(4.60) cn(t) = 0 for t E K o, n = 1,2, ...

and if t ¢ Ko by:

cdt) = ~I n-I

{:. if t E U K k , n ~ 2

(4.61) k=1

cn(t) = n-I

if t ¢ U K k • k=1


Then consider t ¢ Ko; thus t E Kp for some p ~ 1 and the sequence {en(t)} is none other than (~1' ~2'" ., ~p, 0, ... ,0 ... ) which is total and orthonormal in Jf'(t) = Xn(t) = Xp. We then put:

(4.62) Jf' = fEll Jf'(t)dl1(t), Jf'(t) defined by (4.55) .

We now establish

Proposition 6. With the hypotheses of Theorem 2, there exists an isometry "If"" of if (defined in (4.42» onto Jf' (defined in (4.62» which isometrically maps D( C) (given the graph norm) onto the space of vector fields

(4.63)

given the norm

(4.64)

Moreover

(4.65)

Proof We only indicate how the unitary operator "If"" is defined, the rest being easily verified.

Let

Let N be a fixed integer and t E K N ; then from the definition of K N , there exist N integers:

pf(t) < p~(t) < ... < pZ(t)

with t E Fpf(t) n ... n FpZ(t)"

Put

(4.66) ff(t) = ~:(t) (t), k = 1, ... ,N

and

(4.67)

where

(4.68) f(t) = {J/of(t)~k for

for

we then have for t E K N , N E N*

N

t E K N , N = 1,2, ...

t E Ko;

(4.69) If(t)l; = L Iff(tW k=l

135 See C and D( C) in (4.43).


and consequently

(4.70) L If(t)l~dJl(t) = N~l tN(Jl Iff(tW)dJl(t) = lila·

We leave the proof of the other properties to the reader. o Bringing together the result stated in Proposition 6 and the results obtained in Theorem 4 and Proposition 2, and putting.

(4.71)

we have proved

Theorem 4 (The von Neumann-Dixmier diagonalisation theorem). Let A be a self-adjoint operator with domain D(A) dense in the separable Hilbert space H. There exists:

i) a Hilbert integral Yf = f.l Yf(A)dJl(A), dJl(A) a bounded Radon measure ~ 0 on IR; ii) a unitary operator U from H onto Yf which maps D(A) onto Yfl where

(4.72)

and such that

(4.73) U(Ay) = A. U(y) for all y E D(A) .

Remark 3. A situation particularly important for applications is one where A satisfies (4.34).

(i.e. (4.34) (Au, u) ~ IXlul 2 for all u E D(A), IX constant> 0) .

In this case, the support of the measure dJl is contained in ] Ao, + 00 [ where O<Ao<lX. We then have

Corollary 3. If A satisfies the hypotheses of Theorem 4 and also (4.34), then the Hilbert integral is given by

f+OO

Yf = Yf(A)dJl(A) . ;'0

Further, the unitary operator U satisfies for 0 ,,; () ,,; 1:

(4.74)

(4.75)

with

(4.76)

U is an isomorphism from D( A 0) onto Yfo where

Yfo = {u E Yf; AOU E Yf}


Proof Immediate, beginning with the Corollary 2 to Theorem 2, and Proposition 5. 0

We note that u E £e (0 < e ~ 1) if and only if AeU E £ (because Ao > 0 and e > 0) and the norm lule given by (4.76) on £e is equivalent to the graph norm.

4. An Application: The Intermediate Derivative and Trace Theorems136

Here we return to the situation considered in §3.6. Let X, Y be two complex Hilbert spaces. We assume:

{X q Y

(4.77) X is dense in Y.

We again denote by [X, Y]e, 0 < e < 1, the intermediate space between X and Y defined in §3.6, and we consider the space

(4.78) w(m)(a, b) -= {u E L2(a, b; X); ~:~ E L2(a, b; Y)} where (a, b) £; IR and m is a positive integer. Then, given the norm

(4.79)

w(m)(a, b) is a Hilbert space in which f0(]a, bE; X) is dense. For the proof of these two properties, the reader should refer to Chap. XVIII. We can now prove

Theorem 5 (The intermediate derivative theorem). Let u E w(m)( a, b), m ~ 2. Then

(4.80) u(j) E e(a, b; X j ), Xj = [X, Y]i/m, 1 ~ j ~ m - l,j integer,

and the mapping u ~ u(j) is continuous w(m)(a, b) -> L2(a, b; X j ).137

Proof 1) Thanks to the extension technique, for which one should refer to Chap. XVIII, it suffices to prove the theorem for (a, b) = IR . We then use the Fourier transformation in t. If E is a Hilbert space, then

• ~. 1 1 ', u ~ u = .YI' u, u(r) = -- e- I tu(t)dt fo ~

is a unitary isomorphism e(lRt ; E) -> L2(lRt; E) . Then (see vector valued distributions in Chap. XVI)

(4.81) u E w(m)(IR) ¢> {u E. L2( lRt; X) rmu E L2(lRt; Y) ,

136 The results of this section can also be found in Lions-Magenes [Il 137 u(j) denotes the j-th derivative of u.


so that (4.80) is equivalent to

(4.82) riu E L2(lRt; Xi)' Xi = [X, Y]i/m, 1 ~ j ~ m - 1 .

2) We then introduce the spectral decomposition of the self-adjoint operator A defined in §3.6 and make use of Corollary 3.

{ U (defined in Theorem 4) is an isomorphism of L2(1R; [X, Y]8)

(4.83) onto L2(1R; Jt"1-S) ,

and (4.84) u E w(m)(lRt ) <0> {U(U) : L2( lRt; Jt"1)

,mu(u) E L2(lRt; Jt")

(Jt"8 given by (4.75)). If we put v = U(u)( = v(A., r)), then (4.84) is equivalent to

(4.85) r f+oo (A.2 + ,2m)lv(A., ')I~dJL(A.)d, < + ex)

JIR "0 i.e.; (A. + 1'lm)v(A., ,) E L2(lRt; Jt")

and the property to be proved is equivalent to

(4.86) A. 1 - i/mlrl i v(A., ,) E L2( lRt; Jt") ,138

with

(4.87) {fIR 1: 00 A.2(1-i/m)lrI2ilv(A.")I~dJL(A.)dr ~ CII(A. + Irlm)vlli'(IR,;Jf")

C = constant> O.

From the Holder-Minkowski inequality we have:

(4.88) P p' {

A.l-i/mlrli ~ ! A.(1-i/m)p + ~ 1,lip'

11 k, - + --, = 1, and ta mg p such that p p

(1 -j/m)p = 1, jp' = m,

whence we deduce (4.87) and Theorem 5. o We now go to show that the functions u(j) E L2(1R; X) are in fact, after modification on a set of measure zero, equal to functions continuous in larger spaces. To this end, E being a Hilbert space, we introduce the space:

{

<c°([a, b]; E) = the space of functions continuous .?J(a b' E) = on [a, b] with values in E if a, bE IR, or

, , the space of bounded continuous functions from (a, b) -+ IR if a or b or a and b are infinite;

138 The notation 1 - jim signifies 1 - {, or again 1 - (jIm). m


~(a, b; E) is given the norm:

IIcplloo = sup Icp(t)l· IE[a, b]

Theorem 6. The mapping u H u(jl of.@( [a, b]; X) into itself extends by continuity to w(ml(a, b) - 88(a, b; Xe) where

)

(4.89) Xe). ~ [X Y]e e. = j + 1/2 0 ~ J' ~ m - 1 . , j' J m'

Remark 4. Since def j + 1/2

Xi = [X, Y]jlm ~ [X, Y]ej' ej = m '

if u E w(m)(a, b), then u(j) is square summable with values in Xj and continuous in a larger space. 0

Proof We reduce it to the case where a = - 00, b = + 00 (see Theorem 5); we again use the Fourier transformation and the diagonalisation of the operator A furnished by Theorem 4. i) For u E .@(IR; X), we put v = U(u), and note that we then have:

(4.90)

which is equivalent to

e. = j + 1/2 J ' m

with

(4.91) IIrjvIlL'(IR,;Jt',_.) ~ c.llull m;

( (4.91) follows in effect from

. Irli}.I-8j IrIJ}.I-ej v = (Irl m + }.)v

Irlm + ). ,

whence, using the Cauchy-Schwarz inequality and the change of variable r = }.llmO',

r ( r [ 100I i J2 )112 ) JlRllrlli}.l-ejv(A,r)IAdr~ JIR 1+IO'lm dO' ·lIull m ·

ii) Since the Fourier transformation ff or ff -I maps Ll(lR; E) into ~(IR; E) then, from (4.90)-(4.91). ff -1 « ir)j v) is in particular in 88( IR; £1 -e), and u(j) = V-I (ff - 1 «ir)j v)) is in ~(IR; X e), the mapping u H u(j) being continuous on .@(IR; X) given the topology of w(m{(IR) in ~(IR; Xe). 0

)

We now prove the following trace theorem:


Theorem 7. Let u E w(m)(o, +oo);from Theorem 6 we have:

() . =j + 1/2 } .

m

Then the mapping: m-l

(4.92) u H {u(j)(O) , 0 ~ j ~ m - I} of w(m)(o, +(0) -+ n X fJj

j=O

is surjective.

Proof 1) We first remark that to prove the surjectivity of (4.92) it is sufficient to verify the surjectivity of U H uU)(O) from w(m) (0, + (0) -+ X 8j for any fixed j, O~j~m-1.

m-l

In effect, assuming the surjectivity for fixed j, if {aj } E n X/Jj' then there exists j=O

Uj E w(m) (0, + (0) with u}j)(O) = aj •

der m Let Uj(t) = L ckjuj(kt), where the Ckj are defined by:

k=1

(4.93) m {O if p # j, ° ~ p ~ m - 1 ; L kPCkj = 1 'f k= 1 1 P = j;

we have Uj E w<m)(o, + (0) with

(4.94) U(P)(O) = {O ~f } a· If

}

m-1

p#j, O~p~m-l p = j.

Then IJIt = L Uj E w(m)(o, + (0), IJItU)(O) = aj' j = 0, ... , m - 1; IJIt thus j=O

m-1 defined is called a lifting of {aj } E n X 8j in w(m) (0, + (0).

j=O

2) It remains to show that we can define a lifting for fixedj, 0 ~ j ~ m - 1. So let aj E Jf'1-8j; it suffices to construct

{Wj EL2(0, +oo;Jf'd, w}m)EL2(0, +00;Jf')

(4.95) wV)(O) = a. J J '

with

(4.96) { II Wj II £'(0, + oo;.It'd + II w}m) II L2(0, + 00;Jt") ~ C .11 aj II Jt".) ; then u = U -1 (Wj) will give u E w<m) (0, + (0) with u(j)(O) = aj' whence the theorem. 3) To construct Wj we consider qJ E ~([O, + 00 [) with qJ(i)(O) = 1 and define Wj

by:

(4.97) wj (-1., t) = r1/maj(-1.)qJ(-1.1/2t) ;

it is easily verified that we then have (4.95) and (4.96). o From the trace theorem it follows that the spaces [X, YJ8 have the interpolation property given after Definition 8, §3.


Remark 5. We have:

(4.98) (J . = j + 1/2 J ' m

Vu E w(m)(o, 00) .

In effect, p being a parameter > 0, let

up(t) = p-ju(pt). Then u ~j) (0) = u(j) (0). Furthermore, from the continuity of u ~ uU)(O), 3cj > 0 with:

(4.99) lu(j)(0)lx8, ~ cj(lulu(o. +co;X) + lu(m)IL2(o. +00;Y))'

Applying (4.99) to up and choosing p such that (u being fixed)

p-j-tlulu(o. +oo;X) = pm- j - t l u(m)IL2(O. +00;Y);

we obtain (4.98) with Yj = 2cj .

175

o Remark 6. The space X 9. is called the "trace space" ; in effect it is generated by the

. J

"traces" u(})(O) of the u describing w(m)(o, + 00). It is then natural (see Theorem 7) to introduce on X 9j the quotient norm:

(4.100) Ill a lll x8, = inf { II u II wm(o. oo)}

UE w~~! +~) uW(O) = a

it is easily verified that the norm (4.100) is equivalent to the initial norm o

5. Generalised Eigenvectors

5.1. Introduction and Examples

Let H be a separable, complex Hilbert space; we denote by ( , ) the scalar product in H. Consider a self-adjoint operator A on H with compact inverse; we have seen that the eigenvectors (WdkEN form an orthonormal basis for the space H, such that every element x E H can be represented in the form:

+00

(4.101) X = L XkWk ' X k = (x,wk )· k=O

In the case where A is an arbitrary self-adjoint operator, the formula:

(4.102) x = f:: dE;.(x) (E;., A E IR, spectral family of A)

is a generalisation of (4.101). We note that a more natural generalisation of(4.101) would be a representation of the form:

(4.103) f+oo

X = -00 w;.dp(A)


where WA would be an element of H satisfying the equation

(4.104) AWA = AWA

and where the measure dp(A) would play the role of the coefficients Xk of the formula (4.101). However, if one considers examples of the simplest unbounded self-adjoint operators or of unitary operators which have a continuous spectrum, one observes that these operators have no eigenvectors in the Hilbert space H.

Example 4. Let H = L2(1R) and let A be the operator defined by

(4.105)

We noted above139 that this operator (corresponding in quantum mechanics to the position operator) has a continuous spectrum. If we try to solve in L2(1R) the equation

(4.106) Xh. = Ah. , we see that the only possible solution is

(4.107) { h. = 0 X.# A h. (A) arbitrary.

In the space L2(1R), one function possessing the properties (4.107) is the null function. But we can try to solve (4.106) in a larger space, for example .@'(IR). Then, in the distribution sense (i.e. in .@'(IR», equation (4.106) admits these solutions:

(4.108) h. = c<5 A , (<5 A = <5(x - A», c constant.

The Dirac measure at a point x = A is a distribution which does not belong to L 2(1R); it is a generalised eigenvector of A. Recall that the spectral family {E A} associated with A is such that

(4.109) { l if x~A Ed = XA·f, VfE H, XA(X) = 0 l'f

X > A,

so that in the distribution sense we have:

(4.110)

Example 5. We now consider the group of translations {1h} in the space H = L2(1R). Suppose thatfbe such that

def (4.111) (7;.f)(t) = f(t - h) = Af(t) , VfE H .

139 See §3, Example 1.

§4, Hilbert Sum and Hilbert Integral 177

Introduce the Fourier transformation !7 in L 2 (IR) defined by

A A 1 f+oo 'I (4.112) !7u = U, u(r) = rc u(t)e-It dt v 2n - 00

which is an isometric isomorphism of L2(lRt) onto L2(lRt)' Then !7(Thu) = e-ithu(r), so that equation (4.111) is equivalent to

(4.113)

which implies that l( r) must be null for all r E IR such that e - ith # A. Thus j( r) = 0 except perhaps on a countable set of points. Since such a set has Lebesgue measure zero, j = 0 a.e. whence f = O. It follows that the operator Th has no eigenvectors in the space H. However, we can find functions which do not belong to H and which are eigenvectors of Th , for example the W;. defined by

(4.114)

satisfy:

(4.115)

W;.(t) = e+ ial

In virtue of the Fourier formulas, we have:

f(t) = -- f(a)e+latda, f(a) = -- f(t)e-la1dt 1 f+OO A , A 1 f+oo , fo -00 fo -00

and:

Thf(t) = -- e,ahf(a)ela1da = - e,ah f(x)e-laXdx ela1da. 1 f+oo ,A, 1 f+oo . [f+oo 'J ' fo -00 2n - 00 - 00

Hence we can write

i) f = f:: (f, W;.)W;. dp(A), dp(A) = 21n da(A)

(4.116)

entirely analogous to the formula (4.101), and Ax = L AkXkWk • k

The system of eigenfunctions given by (4.114) and (4.115) will then be said to be complete because of the Plancherel-Parseval formula:

(4.117)

which is here written:

(4.118) f+OO Ifl2 = _ 00 l(f, w;.W dp(A) .

We see that the unitary operator Th has no eigenvectors in the space L2(1R), but


nevertheless does have a complete system of eigenvectors (which are generalised eigenvectors) outside the space L 2 (IR). Such a situation holds again for the self-adjoint operator defined in L 2 (IR) by

A = - i ddx ' which commutes with the translations T,,; it is easy to see that the w;.

defined by (4.114)-(4.115) again form a complete system of generalised eigenvectors

f d. 2() or A = - i dx m L IR.

5.2. Definition of a Functional Framework

Consideration of the previous examples suggests that if we seek a representation analogous to (4.103) then we should envisage developments which arise out of the space H. The principle by which we realise this objective is then the following. We construct a topological vector space cP contained with continuous injection in the space H and dense in H. Then we identify H with its dual H' (in fact, its anti-dual since H is complex), and can consider H as embedded in its dual (in fact, anti-dual) cP' of cPo We thus have the situation

(4.119) cP c; H c; cP'

each space being dense in the following one. A triplet space (cP, H, cP') where H is a Hilbert space and cP a nuclear, countably hilbertian space140 satisfying (4.119) is referred to in the literature as a "rigged" Hilbert space141 ; A being self-adjoint, the construction (4.119) is realised so that

(4.120) A(cP)ccP.

Then using an adequate definition of A in cP', cP' contains the eigenvectors of A, which, if they do not belong to H, are called generalised eigenvectors of A. Recalling that a topological vector space cP is countably hilbertian if its topology can be defined by means of a countable sequence of hilbert norms {II II n }nE F\J *, such that if cP n denotes the completion of cP for the norm II II n' we have: cP n + 1 c; cP nand

(4.121) n E F\J*

cP is then, in particular, a Frechet space (i.e. locally convex, metrisable and complete), and a metric on cP can be defined by

(4.122) 00 1 II u - v II n

d(u, v) = nf:1 2n-1 1 + lIu - vlln .

140 See Definition 6 below. 141 Or an "Equipped" Hilbert Space. In the french translation of Gelfand-Vilenkin [1], such a triplet is called a "triade hilbertienne".


If <P~ denotes the dual (in fact, the anti-dual) of <Pn which is a Hilbert space, <P~ is itself a Hilbert space for the norm

(4.123) IlfII-n = sup I<f, u>l, fE <P~, II u II. ,;;; 1

the <, > denoting the duality (in fact, the anti-duality) between <P~ and <Pn. Thus we have:

(4.124) { <P ~ ... ~ <Pn+ 1 ~ <Pn ~ ... ~ <P1 <P'1 ~ ... ~ <P~ ~ ... ~ <P'

and

(4.125) <P'

This allows us to pose the

Definition 6. A countably hilbertian space <P is nuclear if for all mEN *, there exists n E N * with n ~ m such that the injection of <Pn into <Pm is a nuclear operator.

Remark 7. If <P is a Hilbert space then it is countably hilbertian: its topology is defined by a single hilbert norm. It then follows from Definition 6 that a Hilbert space is nuclear if and only if the identity is a nuclear operator and hence compact. Consequently, a nuclear Hilbert space is finite dimensional. 0

Remark 8. Let <P be a nuclear countably hilbertian space. Let in. m be the injection of <Pn into <Pm. If this injection is nuclear it can be written:

00

(4.126) in. m( <p) = L Ak( <p, <Pk)nl/lk' <P E <Pn k;l

[where ( , )n is the scalar product in <Pn]; and {<p p }, {I/Id are orthonormal bases for 00

<Pn and <Pm respectively; Ak ~ 0 and L Ak < + 00. o k;l

Remark 9. More generally, a space F for which the topology is defined by a family of norms Pn is nuclear if there exists a system of equivalent norms p~, such that: i) the completion <Pj of <P for pj is a Hilbert space;

ii) for each m, there exists n such that p~( <p) ~ p~( <p), for all <P E <P and the injection of <Pn ~ <Pm is a Hilbert-Schmidt operator. It can then be shown (Grothendieck [1]) that: 1 ° each finite dimensional space is nuclear;

00

2° each direct sum EB <Pk, of nuclear spaces <Pk is nuclear; k;l

3° if {<Pk)kEN* is a sequence of nuclear spaces <Pk, then the strict inductive limit of the <Pk, denoted by lim <Pk, is again nuclear:

--+

the topology induced on <Pm by <Pn being identical to that on <Pm; 4° if <P is a nuclear Frechet space then its dual <P' is nuclear. o


5.3. Construction of a Triplet (t/). A. t/)")

The operator A being self-adjoint in H with domain D(A) dense in H, a natural way of constructing a countably hilbertian Frechet space satisfying (4.119) and (4.120) is the following: We denote by D(A k) the domain of A \ kEN * and let

(4.127)

(4.128)

tP = n D(Ak) (AO = I, D(AO) = H); kEN

k = 0,1, ...

(where I I is the norm in H) which makes D( A k), kEN a Hilbert space. Hence tP is endowed with the structure of a countably hilbert ian Frechet space. 142

Note that tP is dense in H, since if {E;,} denotes the spectral family of A, then tP contains all the vectors:

(4.129) E~(u) = J: dE;,(u), u E H, a < b, a, bE IR ,

which form a dense set in H. Let us identify H and its dual H' (in fact, its anti-dual). Then the duality < , > (in fact, anti-duality) between H' and H coincides with the scalar product on H. It can be shown, by using the topological structure of the dual of a Frechet space and the Hahn-Banach theorem, that one has:

(4.130) {if T E tP', there pexists pEN *

such that T = L Ai T; , i=O

and T; E H, 0 ~ i ~ P ,

the representation (4.130) not being unique. It is thus clear that tP so constructed satisfies (4.119)-(4.120); but such a tP is not in general nuclear, as the following example shows:

Example 6. We take

(4.131) { . d

H = L2(1R) , A = -I - with domain dx

D(A) = {u;uEL2(1R), u'EL2 (1R)} = Hl(IR).

142 <P is a locally convex topological vector space with a countable neighbourhood base, and is metrisable: one can always equip <P with the metric

1 P (u - v) d(u, v) = I ___ k __ -

k 2k I + Pk(U - v)

and the fact that <P is complete results from the fact that A is closed.


Then

(4.132)

and

(4.133) +00

cP = n Hk(lR) = !?fiu .143 k=O

181

Thus (Schwartz [2], vol. 2, p. 57) !?fi u and !?fiLl are not Montel spaces, that is to say such that their bounded sets are relatively compact. Thus if cP is nuclear, its dual CP' is a Montel space (Gelfand-Vilenkin [1]). Thus cP defined by (4.133) is not nuclear. 144

However, we observe that in this case the space 9"(IR) of rapidly decreasing functions satisfies:

(4.134)

(where ~ denotes the continuous injection with dense image) and the space 9"(IR) is itself a nuclear space such that the triplet (9"(IR), [2 (IR), .'l"(IR» forms a "rigged space" in the sense defined above in §4.5.2. In the general case we can demonstrate

Proposition 7. Let A be a self-adjoint operator in H, with domain D(A) dense in H. Let cP be defined by (4.127)-(4.128). Then one can construct a nuclear space 'I' ~ CP, dense in H, which satisfies (4.120).145

Proof The space cP being dense in H and separable, there exists a sequence {Un}nEN* dense in H, with Un E CP, Vn. Let VI = U I and let

(4.135)

CPI.k is a vector space of finite dimension, thus nuclear. Let CPI = ~ CPI,k, CPI is nuclear, CPI ~ CP, A( CPt> C CPl' If CPI is dense in H, we take 'I' = CPl' If CPI is not dense in H, there exists V2 E {U2' U3, ... }, V2 not belonging to the closure of CPI for the norm in H; we consider

(4.136) CP2,k = {v2,Av2,·· .,Akv2} and CP2 = ~ CP2,k'

We then form CPI EEl CP2 which is nuclear and which, ifit is dense in H, can be taken for '1'. If not, we continue, and then construct

(4.137)

which answers the question. o 143 Where flfi L , is the space of functions «foo which are in U together with all their derivatives. 144 Note that a nuclear countably hilbertian space is itself a Montel space (see Treves [I], p. 159) and that its strong dual is nuclear. 145 One can prove (see Triebel [I], p. 478) that 4> is a Montel space if and only if A has a pure point spectrum (for 4> to be nuclear, a supplementary condition on the eigenvalues of A is needed; see Triebe! [I]).


5.4. Generalised Spectrum

Being given a triplet (<P, H, <P') satisfying (4.119) and (4.120) (<P not necessarily nuclear), we can then define the action of A in <P' (that is to say extend A to <P') by posing, by definition:

(4.138) {if TE <P',

AT( <p) = < AT, <p) = < T, A <p ) for all <p E <P .

Definition 7. A generalised eigenvector of A, w)" is then a continuous linear form on <P, thus w), E <P', satisfying:

(4.139) <w;.,A<p) = A<W),,<p) forall qJE<P.

Remark 10. The space ~(IR) is a nuclear Frechet space. Its dual ~'(IR) is a space of distributions with compact support and is nuclear. Hence 15), E ~'(IR) is in the sense of(4.139) a generalised eigenvector of the operator A of multiplication by x in L2(1R) (see Example 4 above). 0

Remark 11. In an analogous manner, 9'(IR) is a nuclear Frechet space. The eigenfunctions e - ilx form the generalised eigenvectors of the translation operator Th 146 (Example 5); in effect, for all A E IR, e- ilx defines a tempered distribution (i.e. <p E 9" (IR)). We note here that the value at A of the Fourier transformation of the function qJ E 9'(IR), cPU), say, is none other than the value of the form e- ilx on the function <p:

cP(A) = <e- ilx, <p) .

The Plancherel-Parseval equality (4.117) shows that the set of generalised eigenvectors {e - ilx} is complete, in the sense that <p = 0 if cP = O. 0

Remark 12. Regarding the idea of giving an analogous representation to the formulas (4.116) in the case of a self-adjoint operator A (or a unitary operator) in H, we remark that if <P is a nuclear, countably hilbertian space, such that (<P, H, <P') satisfies (4.119)-(4.120), then the injection i: <P -+ H is nuclear. It is useful for what follows to have a representation for this injection i. 0

We now prove

Proposition 8. Let <P be nuclear and such that (<P, H, <P') satisfies (4.119). Let i be the canonical injection of<P -+ H. Then there exists n E N, an orthonormal basis {ed of H and an orthonormal basis {xd of <P~ such that

00

(4.140) i(qJ) = L Ak<Xk' qJ)ek , for all <p E <P k=1

with

(4.141)

Proof Denote by ik the extension of i to the space <Pk ; i being nuclear, there exists n E N, such that in the injection of <Pn into H is nuclear. Then we know (see §2) that

146 Which is unitary.


we can find an orthonormal basis {ek}, (resp. {cpd) of H (resp. «1>n) such that for cP E «1>n we have:

c»

(4.142) in(CP) = L Ak(cp, CPk)nek k=l

with the Ak satisfying (4.141) (where ( , ) is the scalar product in «1>n); if cP E «1> then in(CP) = i(cp) by definition of the in. Thus we have:

c»

(4.143) i( cp) = L Ad cP, CPk)nek for all cP E «1> • k=l

Now, for fixed k, cP H (cp, CPdn is a continuous linear form on «1>n; hence there exists Xk E «1>~ such that

(4.144)

and Proposition 8 follows. o We now introduce the spectral decomposition of the self-adjoint A, and denote by O/i the isometry which maps the space H onto the Hilbert integral

where d/1 is a measure > O. We also use the notation:

.Yt = f' .Yt(A.)d/1(A)

(4.145) cjJ: A H cjJ(A) = (O/iCP)(A) , cP E H .

We can now demonstrate

Proposition 9. Let (<<1>, H, «1>') be a triplet satisfying (4.l19) associated with a selfadjoint operator A, with «1> nuclear and such that (4.120) holds. Then for all A E A = u(A), there exists a nuclear operator ik which maps «1> into .Yt ( A) and we have:

(4.146) { for all cP E «1> ,

cjJ(A) = i,,(cp)d/1 a.e. A,

where c,O(A) = (O/icp)(A).

Proof From Proposition 8 there exists n E N, such that the injection i of «1> into H can be written:

c»

(4.l47) i ( cp) = L Ak < Xk' cP > e k' cP E «1> k=l

where {ed is an orthonormal basis in H with elements in «1>, Xk is defined by (4.l44) (with Xk E «1>~), and with (4.141). For cP E «1> c:; H, we have fp = O/i(i(cp». We now show that

(4.148)

ik then being nuclear from «1> into .Yt(A.).


00

Let us show that the series L Ak <Xk' IP> t\(A) is absolutely convergent k=l

(a.e. relative to dJ.l) in Jr(A). Since {ed is an orthonormal basis in H, {ed is an orthonormal basis III

Jr = f' Jr(A)dJ.l(A). Thus we have

(4.149) 1 = lekl 2 = r lek(AWdJ.l(A) , k = 1, ... J2EA

Note that for all N EN,

00 00

Since the series LAk is convergent, the convergence (a.e.) of LAklek(A)I", can be 1 1

00

deduced ifLAklek(A)li converges. But this follows (a.e.) from 1

Now,

and since II Xk II <1>; = 1, k = 1,2, .. : N +00

LAkl <Xb IP> II ek(A)I" ~ II IP II •. L Aklek(A)I" , dJ.l-a.e. A, 1 1

00

whence it follows that we can define L Ak <Xk' IP> ek(A) a.e. with respect to dJ.l and k=l

that we do have (4.148). We can also deduce that:

00 +00

(4.150) li"(IP)I,, ~ L Akl <Xk' IP> Ilek(A)I" ~ II IP II.· L Aklek(A)I" dJ.l-a.e. k=l 1

which establishes the continuity of i,,: tP 1--+ Jr, defined (a.e. A) by (4.148). (Naturally, i" can be defined for all A E A as a nuclear operator from tP into Jr(A) by taking i" = 0 for each ofthe points A where i). is not defined by (4.148)). Whence Proposition 9. 0

We can now demonstrate

Theorem 8. Let A be a self-adjoint operator in the (separable) Hilbert space Hand let tP G H G tP' with tP nuclear satisfying A tP c tP. Then A admits a complete 147 system of generalised eigenvectors.

147 In the sense that i;.(q» = 0 a.e. implies q> = 0 in 4>.


Proof Consider a Hilbert integral .tf = fll .tf (.A.) d~(.A.) associated with A.

For all cp E CP, we have t7>(.A.) = i;.(cp)(i;. defined in Proposition 9). It follows that to each element ~ E .tf(A) there corresponds an element [ E CP', defined by:

(4.151)

Recall that A is defined on cP' by:

<A[,cp) = <[,Acp).

Hence the isometry iJ/I makes kp(A) = A~(A) correspond to Acp. From (4.151) we have

Thus

{the form [ corresponding to ~ E .tf(A.) is a generalised

(4.152) eigenvector of A .

If now (<p(A), ~h = 0 a.e. A, then

t7>(A) = 0 a.e. implies Icpl2 = ilt7>(A.)lid~(A) = 0,

therefore the set of generalised eigenvectors is complete. Whence the theorem. o Remark 13. Theorem 8 is still true whenever cP is not nuclear but satisfies (4.119)-(4.120), if the injection of cP into H is nuclear. 0

Remark 14. If A is a unitary operator, Theorem 8 still holds for A (it reduces to the self-adjoint case via Cayley transforms). 0

Remark 15. Let {Ai' A 2 , ••• , AN} be a system of N self-adjoint operators whose resolvants commute. (The system is said to commute in the resolvant sense.) Then it is possible148 to construct a triplet cP G H G cP' satisfying the hypotheses of Theorem 8, with cP nuclear. Then we have

AiAkCP = AkAiCP for all cp E CP, i, k E {l, ... , N} .

A linear form X E cP' will be said to be a generalised eigenvector of the system {Ak} if one has:

(4.153)

the set A = {Ai' ... , AN} will be called the eigenvalue corresponding to the generalised eigenvector X . 0

148 See Maurin [1] .


We thus have

Theorem 9. If {Ak }, i :( k :( n is a system of commuting self-adjoint operators in the rigged Hilbert space (cP, H, cP'), the set of generalised eigenvectors is complete. 0

There is another route by which we may construct generalised eigenvectors (Gelfand-Shilov [1]) . Let cP be a countably hilbertian space. A family of continuous linear forms {X)'}'.\E[a.b] E cP' is said to have strongly bounded variation on [a, b] if there exists pEN such that X). E cP~ for all A and for all partitions of the interval ] a, b] into intervals ] Aj , Aj + 1] where a = Ao < A1 < ... < An = b, one has:

n-1 (4.154) " II x). + I - x). II A.' < c . ~ J J 'Yp

j=O

It can then be shown (see Gelfand-Shilov [1], Chap. IV, Theorem 2) that, for separable cP and an arbitrary positive measure dll, if the family x). has strongly bounded variation in A then one can find pEN *, w). E cP~ such that

(4.155) lim <XHh - X).,q» = <w).,q» forall q>EcP. h~O 1l(]A + h,A[)

We then put

(4.156) w). = ~~ (the weak derivative of X;. with respect to Il) .

In the Hilbert space H consider the spectral family {E).} attached to A and a fixed vector g E H. Then the family x). defined by

(4.157) <x)., q» = (E).g, q», Vq> E H ,

does not have strongly bounded variation in A on H. Nevertheless, it has weakly bounded variation in A: that is to say, for every subdivision { Aj} of the A-axis into a finite number of intervals .1 j = [A j , Aj + 1] we have

(4.158) 1

~1((E).j+ I - E).j)g, q»I:( "2 (lgl 2 + 1q>12) = c(q» , Vq> E H .

The problem then posed is whether the X). defined by (4.157) is differentiable with respect to a measure >0, Il. If the space cP is nuclear, every X). E cP' with weakly bounded variation on cP also has strongly bounded variation. If the spectrum of A is simple, it is known that there exists a cyclic vector g (a generator element for H); then it can be shown (Gelfand-Shilov [1]) that

(4.159)


defines on rJ> a continuous linear form given by:

(4.160) d --

<W .. , qJ) = -d (E .. g, qJ), for all qJ E rJ>, fJ ..

which is a generalised eigenvector of A. We then have the representation

(4.161)

187

It is this formula (4.161) ii) which establishes that the generalised eigenvectors form a complete system, because one cannot give any meaning to <X;., x .. ,), A "# A' . In the case where the spectrum of A is not simple, then H, as is known, is the direct Hilbert sum of cyclic subspaces Hn:

00 00

H = EB Hn ~ EB L;n· n=l n=l

The completeness relationship can be written

(4.162) IqJI2 = J1 f:: l<wl ,qJ)1 2dPn(A),

dPn(A) = d(E .. Wn, wn ) where {wn } is an orthonormal basis for H, and

< wl , qJ) = dP~(A.) (E). Wn, qJ),

and qJ is represented by:

(4.163) n~l f:: <Wl' qJ )wldPn(A).

These two formulas (4.162), (4.163) are the translations here of the formula (4.146).

Appendix. "Krein-Rutman Theorem" *

Krein and Rutman have generalised to a class of positive operators (in a sense to be made precise) a result well known in the theory of matrices, and due to PerronFrobenius. If M is an n x n matrix with strictly positive coefficients, then M has a positive eigenvalue, A, larger in absolute value than all the others (thus A = p(M), the spectral radius of M) and the components of the associated eigenvector all have the same sign (thus for example positive). Let X be a real Banach space with norm denoted by II II x.


Definition 1. We say that a closed set in X is a cone in X ifit possesses thefollowing properties: 149

i) ° E K, ii) u, v E K => au + flv E K Va, fl > 0,

iii) v E K and - v E K => v = 0.

Definition 2. A cone K c X is said to be reproducing if X K - K.

Example. Let Q be an open set in IR" and X = U(Q), 1 ~ p < + 00;

K = {u E U(Q) , u > ° a.e. in Q}

is a reproducing cone since u = u+ - u- (u+ is the positive part and u- is the negative part of u).

Definition 3. If K is a cone in X, then a set K * c X * is said to be a dual cone if

<f*, v) > ° Vv E K .

For example, if K is the cone of positive functions in e(Q), then K * K.

Definition 4. Given u, v E K, we say that u ~ v if v - U E K. In particular, all the elements of K satisfy v > 0. A cone is said to be normal if

u, V E K, u ~ v => II u II x ~ II v II x .

Definition 5. Let B E ~(X). If B leaves invariant K (B(K) c K) then B is said to be a positive operator. We now let K be a cone with non-empty interior k. Definition 7. Let B E ~(X); B is said to be strongly positive if Vv E K, v #- 0, Bv E k. Theorem 1. -Ist Krein-Rutman Theorem (The strong theorem). Let K be a reproducing cone, with interior k #- 0, and let B be a strongly positive compact operator on K. Then the spectral radius of B, r(B), is a simple eigenvalue of Band B*, and their associated eigenvectors belong to k and k * (more precisely, there exists a unique associated eigenvector in k (resp. k *) of norm = 1). Furthermore, all other eigenvalues are strictly less in absolute value than r(B).

Proof Step 1. First we show that there exists a unique eigenvector of B, Xo E K, II Xo II = 1. In effect, let u E k; then replacing B by wB with w > 0, if need be, we can assume that Bu > u (otherwise nBu - u ¢ K, Vn E N which implies that Bu rt k). Let us introduce:

(1) K, = {x E K; x > W, II x Ilx ~ R}

149 Here we call a cone what in other terminologies is a salient closed cone.

Appendix. "Krein-Rutman Theorem" 189

where R = II B II [1 + ell u Il x]; K, is a bounded closed convex set with ° ¢. K,; we then define:

1 1'.(x) = W B(x + ellxllu).

We observe that 1'.: K, --+ K, so that, from the Schauder fixed point theorem: 150

3x, E K, such that

(2)

Then if xE = -I x, ,we have II xf II = 1, xf E K. Putting Af = ~1_, we can write IXflix Ilxflix

(2) in the form:

(3)

We now further show that

(4)

Thus as e --+ 0, we can extract a subsequence {X., Af} such that AE --+ J1.o E [0, 1] and from (3), B being compact, we have xf --+ Xo E K. Then

Ilxollx = 1, J1.oBxo = Xo, thus J1.o > 0, and

(5) 1

Bxo = - Xo = AoXo . J1.o

Thus the existence of an eigenvector in question is established if (4) holds.

Verification of(4). This reduces to showing that if x E K satisfies

then A ~ 1. Now we have:

thus

x = AB(x + eu) for A ~ ° , x ~ ABx and x ~ AeBu ~ eu ,

x ~ ABx ~ A2 eBu ~ A2 eu;

whence it follows by recursion that x ~ Aneu, and:

1 An X - eu E K "In EN.

If we have A > 1 then whenever n --+ 00, we can deduce that -eu E K, thus - u E K, thus u = 0, a contradiction; whence (4). 0

150 See Berger [1].


We now establish the uniqueness of Xo'

Suppose that there exists Xl i= Xo , Xl E k. such that

since -Xi ¢ K (i = 0, 1), there exists Yi > ° with:

° i= Xi - YjXj E K, i i= j

and such that Xi - AXj E k. if A < Yj and such that Xi - AXj ¢ K if A > Yj. Then

hence we have Yj Ili < Yj' i i= j (i,j = 0,1), so that 110 < 111 and 111 < 110 whence Ilj

uniqueness.

Step 2. We show that Ao = 1/110 is a simple eigenvalue. Let X E X, X ¢ K u (- K) satisfy X = IlBx for 11 E IR. We will then show that 110 < 1111· Since X ¢ K, there exists (jxo(x) > ° such that

Xo + (jxo(x)x E K\{O}, Xo + Ax E k., 0 < A < (jxo(X) ,

and, replacing X by - x, we also have

Thus

hence

Xo - (jxo(x)x E K\ {O}, Xo - Ax E k., A < (jxo( -x)

(jxo( - x) ~ (jxo(x) .

if 11 > 0 we get the first relation above: 110 < 11; if 11 < 0 the second relation gives 110 < - 11 thus

110 < 1111 and dim ker (/ - 1l0B) = 1.

Step 3. We assume that there exists X E X, X ¢ K u (-K) such that

o

IlB(x) = x with 11 = Illleio E C ,

and then show that 1111 > 110' so that 1/110 = r(B). Here we are required to work in the space X = X + iX. We define B in X by B(x + iy) = Bx + iBy.

Appendix. "Krein-Rutman Theorem"

Now let x = x + iy such that

Bx = IlX, Il = Illlei8 ,

and let P be the plane generated by x and y in E. We can deduce that the action of Bin P is given by

B = _1_ ( cos() Sin()) = _1_ R8 IF I III - sin () cos () I III .

191

Since the intersection of the cone K and the plane P reduces to {O}, for every element x E Pj{O} thus rt K, there exists c5(x) > 0 such that

Xo + c5(x)x E K - {O}, Xo + Ax E k, A < b(x) ,

whence it follows by an argument analogous to that made above that we must have:

Ilo ~<

whence the result. The same proof applies to the case of B*. o If B is strongly positive on K, but not compact, there exists an eigenvalue for B* (see Silverman-Yen [1]). On the other hand, we have:

Theorem 2. - 2nd Krein-Rutman Theorem (The weak theorem). Let B be a linear operator positive on a reproducing convex cone K in the Banach space X. If B is compact and has spectral radius r(B) oF 0, then there exists cp E K (cp oF 0) with

Bcp = r(B)cp .

Proof (Krein-Rutman [1]). 1st case: r(B) is an eigenvalue. By putting Ao = r(B) the resolvant R;.(B) of the operator B can be written

+00

R;.(B) = L rk(A - Ao)k (r _ n oF 0) , k = - n

where rb k = (- n, - n + 1, ... ) is a linear operator and where the series converges in the operator norm. We can find u E K such that r_ nU oF 0 [because if LnK = 0 then we will have Lnx = 0 and Ln = 0]. Then

(7) L nU = lim (,l. - Ao)n R;.u . ;. --+;'0

Since Ao = r(B), we have for A > Ao


and since Bnu E K, Vn, then -R;,.u E K (A > Ao) and from (7) we get

(8)

But we also have (B - U)R;,. = 1, BR;,. = AR;,. + 1 ,

whence BR;,.u = AR;,.u + U •

Multiplying this last equation by -(A - Ao)n and passing to the limit as A --+ Ao, we find that

(9) o 2nd case: We assume: 3Ao(IAol = r(B)) an eigenvalue of B such that AO > 0for some n EN. In this case, since Bn K c K, we are back to the first case applied to the operator Bn. Thus there exists v E K with

Bnv = AOV (v #- 0) . We then construct the vector:

v' = IAoln-lv + IA o l n- 2Bv + ... + Bn-lv, v'EK, v' #- 0,

whence

Bv' IAoin - 1 Bv + IAoin - 2 B2v + ... + Bnv = IAoin - 1 Bv

+ ... + IAolnv = IAolv' .

3rd case : No integral power of the eigenvalue A with I A I = r( B) is positive.

o

Let AO be an eigenvalue of B such that IAol = r(B) and has the largest real part. We have Bu = AoU, Ao = p(cos CPo + i sin CPo) where the angle CPo is incommensurable with 2n. Let e > O. Introduce

B. = B + eB2 .

Then the spectrum of B. consists of all A j + eA} with A j E a( B). Suppose IAjl = r(B) = p, Aj = p(coscpj + isincpj); then

Aj + eA.} = p{coscpj + epcos2cpj + i(sincpj + ep sin 2cpj)} ' and

IAj + eA.} I = pJl + e2p2 + 2epcoscpj.

By hypothesis, Re Aj ~ Re AO' Vj, thus cos CPj ~ cos CPo and the eigenvalues of B. with largest absolute value are the complex numbers:

Ao + eA6 and ~o + e~6 . If we now choose e in such a way that the argument (phase) of AO + eA6 is incommensurable with 2n, then from the 2nd case, AO + eA.6 will be a positive eigenvalue of the operator B. with largest absolute value. Since e can be chosen arbitrarily small, from Ao + eA.6 > 0 it follows that Ao > o. Whence the theorem. 0

Appendix_ "Krein-Rutman Theorem" 193

Remark 1. In an analogous way it can be shown that there exists 'I' E K * such that

(10) B* 'I' = )'0'I' .

[To see this we define, in the 1st case, 'I' E K * and we put IjJ = - p,- n'I'. Noting that - L nK c K, it follows that

V x E K, ljJ(x) = < - T!. n 'I', x> = - < 'I', L nX > ;?; 0 ;

thus 'I' E K* and (10) follows by an argument analogous to that used in the proof of formula (5); and similarly for the other cases.] 0

The results established in Theorems 1 and 2 concern the problems of eigenvalues. For non-homogeneous problems we have: under the hypotheses of Theorem 2, and for all ), > reB), the equation

Ax - Bx = y, for each given Y E K\{O}

admits one and only one solution and this solution is positive.

Proof For A > reB), we know (see Chap. VIII) that the equation

h - Bx = Y

admits one and only one solution, and this solution is given by:

x = (I - JiB)- l(JlY) = (I + JiB + ... + Jin Bn + ... )JiY

with Ji = 1/ A, which shows that x is positive.

Under the hypotheses of Theorem 1, the following result can be proved: - for A < reB), the equation

Ax - Bx = Y, for given Y E K\ {OJ ,

admits no positive solution. - for A = reB), the equation

h - Bx = y, for given Y E K\{O} or -y E K\{O},

admits no solution.

o

We now give another version of the theorem which is very useful in the study of examples (especially in neutron theory), the proof of which is due to Sentis (See Sentis [1], Harris [1]). Let (E, C) be a measurable space (the a-algebra Iff being complete); we denote by B(E) the Banach space of bounded C-measurable functions on E (also called universally measurable functions). Let B + be the cone of positive functions. Then we have the

Theorem 3. The Doob--Krein-Rutman Theorem. Let A be a continuous linear operator on B(E) with i) A(B+J c B+; assume further that: ii) there is a positive measure v on E and mEN such that

194

(11)

iii) there exists p > 0 and q> E B + satisfying:

(12)

(12a)

Aq> = pq> ,

{q>(x) "# 0, v-a.e.}.

Then p is a simple eigenvalue of A. Moreover, as n --. 00 then for all f E B(E):

(13)

Proof First we see from (11), (12), (12a) that:

(14) inf q>(x) > 0 . XEE


We can therefore define a positive [from i)] measure on E, by

1 p(l)(x, Y) = --(-) A(q>Iy)(x) where Iy is the characteristic

pq> x (15)

function of Y E g .

This is a probability measure [from iii)]. We now define the iterated probability measures:

(16) p(n)(x, Y) = L p(l)(X, dy)p(n-l)(y, Y), VY E g

I [that pIn) (x, Y) = n--( ) A n( q> 1 y), follows by recurrence], and put:

pq>x

q~ = inf p(n)(x, Y), Q~ = sup p(n)(x, Y) . x x

For all Y E g, we see that:

q} ~ ... ~ q~ ~ ... ~ Q~ ~ ... ~ Q} .

We have to show (below) that for all Y E g, we have:

(17) Q~ - q~ --. 0 if n --. + 00 .

Then the measure qy, the common limit of both q~ and Q~, satisfies:

as n --. 00.

1 p(n)(x, Y) = n--( ) An(q>Iy)(x) --. qy

pq>x

From this we immediately deduce (13).

lSI One can also state this theorem for a linear operator A continuous on L ""(E, v), B+ being replaced by the positive cone of L "'(E, v).


Proof of (17). We note that

(18) p(m)(x, Y) ~ bv(Y) , ( b = inf q>(x) ) pm sup q>(x) .

For (x, y) fixed, the measure tjJx. y defined by

tjJx. y(Y) = p(m)(x, Y) - p(m)(y, Y)

admits a subset Sx. y on which tjJ x.y is a maximum. Let Tx, y = E\Sx, y' Then tjJ x. y is positive on Sx, y and negative on Tx, y and since tjJ(E) = ° we have

Using (18), we obtain:

tjJ x, y(Sx. y) = 1 - p(m) (x, Tx. y) - p(m) (y, Sx. y) ~ I - b [v(Tx. y) + v(Sx, y)]

~ I - bv(E) .

Thus for kEN, we have:

Qtm + m - qtm + m = sup i p(km)(~, Y)[p(m)(x, d~) - p(m)(y, d~)] x, Y E

~ sx~~ [lxy QtmtjJxy(d~) + Lx, qtmtjJx.y(d~) J = suptjJx,y(Sx.y)[Qtm - qtm]

x,y

~ (1 - bv(E»[Qtm - qtm] .

Since (l - b(v(E»k --+ 0, we deduce (17).

Applications of the Krein-Rutman theorems

Example 1. The Perron-Frobenius Theorem. Recall the very simple theorem: If .s::I = (aij) is an n x n matrix satisfying

aij > 0, i, j = 1, 2, . . . , n ,

o

then the linear mapping associated with .s::I is compact and strongly positive and has a simple eigenvalue equal to the spectral radius and an associated eigenvector all of whose components are positive. 0

Example 2. Integral Equations. i) Let a, b E IR1 and E = <'{? 0 ([ a, b] ). We are given a continuous function k: (t, r) I-> k(t, r) strictly positive on the compact square [a, b] x [a, b], and we put

(19) Ku(t) = f k(t, r)u(r)dr, u E E .

Let r + be the cone of positive functions in E.


Since 1'+ i= 0 and since the operator K is then compact and strictly positive, it follows from the 1st Krein-Rutman theorem (strong theorem) that there exists a simple eigenvalue equal to r( K) and an associated positive eigenvector. ii) Let K still be defined by (19) but assume here that E = L 2(] a, b [). Then if r+ = {u E E; u(x) ~ 0 a.e.}, we have 1'+ = 0. From the 2nd Krein-Rutman theorem (weak theorem), there exists an eigenvalue equal to r(K) and an associated positive eigenvector. Here it is not possible to assert that this eigenvalue is simple. However, the function k being continuous and strictly positive on the compact square [a, bJ x [a, bJ, there exists m > 0 with:

(20) k(t,r) ~ m > 0, \ft,r E [a,b];

thus, taking v to be the measure such that dv = mdt, we have:

(21 ) Ku ~ <v,u) = f u(x)dv, UEr+;

the hypotheses (12) and (12a) are then trivially satisfied and Theorem 3 affirms that the eigenvalue r( K) is certainly simple. 0

Example 3. The Laplacian in the Dirichlet Problem. Let Q be a connected regular bounded open domain in fRn with frontier r = 8Q. We consider the Dirichlet problem:

(22) { -Llu = IE e(Q)

u E Hb(Q) ,

which admits one and only one solution u (u E H2(Q) n HbCQ)). We note that if I ~ 0 a.e., then also u ~ 0 a.e. [In effect, u = u+ - u-, u+, u- E Hb(Q) [see Chap. IVJ and if we put

a(u, v) = (grad u, grad v) ,

we have on the one hand:

and on the other: a(u, u-) = - a(u-, u-) ~ 0,

which implies that u- = O.J Denote by r + = {~E e(Q), u(x) ~ 0 a.e.} (1' + = 0). It follows from the preceding that if

B=(-Ll)-1 then

BE2"(e(Q)) and B(r+) c r+.

Moreover, (Q being bounded), we know that B is compact, so that the 2nd Krein-Rutman theorem applies to B, e(Q), and r +; this implies that there exists an eigenvalue of smallest absolute value, and a positive associated eigenvector, for the Laplacian (- Ll) in the Dirichlet problem. We note that the eigenfunctions

Appendix. "Krein-Rutman Theorem·· 197

associated with problem (22) are regular and that it is sufficient to study them in the cone:

(23) P = {u E E = '6'0(Q); u(x)? O} ,

whose interior P consists of functions u > 0; but meanwhile we cannot apply the theorems 1 or 3 directly in the present case because the eigenfunctions we are seeking here have null boundary values. To alleviate this difficulty, we start by determining a function e E H 6(Q) (see Amann [1]) satisfying

(24) { -L1e = 1

eEH6(Q),

and we introduce the space Ee defined by

(25) Ee={UEE; 3A?0 -Ae~u~Ae},

which, given the norm:

Ilull e = inf{A > 0; -Ae ~ u ~ ).e}

is a Banach space. Let Pe = P nEe' P defined by (23) with non-empty interior. Then (see Amann [1]): if B = (- ,1) - 1, If v E P, v i= 0, there exists IXv and f3v (constants with respect to x) such that:

(26)

further, the mapping B is compact from E to Ee. Hence we can now apply the 1st Krein-Rutman theorem to B, Ee , and Pe .

Thus the smallest eigenvalue of the Laplacian (- ,1) in the Dirichlet problem is a simple eigenvalue. 0

Example 4. The Diffusion Multigroup Operator. In the diffusion multigroup problems in neutron theory, we have seen in Chap. lA, §5 that a certain operator A is involved (see Chap. lA, §5, formula (5.37) with X = 1). We shall show that it is possible to apply the 2nd Krein-Rutman theorem to (the inverse of) this operator A under certain hypotheses. We will adopt throughout here the notations of Chap. VII, §2.6, (2.25). Let Q be a connected bounded regular open domain in fR" (in fact n = 3), and introduce the variational framework:

v = (H6(G»G (G E N, G = number of groups chosen), H = (U(Q»G ,

with norms resp.


and the sesquilinear form:

a(u, v) = L Ct1 djgradujgradvj + E (1jk UkVj )dX,

with (real) coefficients dj , (1jk which we suppose "regular" (for example, in ~OO(Q)), and such that there exists a constant (j > ° such that dj(x) ~ (j > 0, "Ix E Q, Vj. We further assume that a(u, v) is coercive:

(28) a(u, u) ~ (jllull~ Vu E V.

Denote by .91 E !e((H6(Q))G, (H- 1(Q))G) the operator defined by a(u, v) (see Chap. VII, §2.6), with inverse B = .91- 1. Then the unbounded operator on H (denoted A) defined by:

{ D(A) = {u E V, du E H} Au = du, Vu E D(A) ,152

has a compact inverse B = A - 1 (this is the restriction of B to H).

Let us show that B conserves the positive cone in H. For each! = {fj}, fj > 0, fj E U(Q), let u = {u j } be the solution in V of the problem:

(29) a(u, v) = (f,v), VVE V.

Then, as in the previous Example 3, decomposing uj into its positive and negative parts: uj = ut - uj-, we get:

hence U-

° ~ -a(u-, u-) = a(u, u-) = (f, u-) ~ ° {u j-} = 0, and the result asserted.

U(A)153

l/r(8) x

We emphasise the fact that we have demonstrated that for each given! = {fj} E H withfj ~ 0, the solution u of the non-homogeneous problem (29) is positive. Hence we can apply the 2nd Krein-Rutman theorem to the operator B.

152 Note that A is a closed operator, but does not necessarily satisfy 'A = A since the matrix (u j .) is not a priori symmetric. 153 In the figures, u(8) and utA) consist of those points which belong to the non-hatched regions of IC.


Thus there exists a positive eigenvector Uo of B for the eigenvalue Ao reB) of. 0:

Buo = r(B)uo . 1

Hence Auo = ~~ uO, and UO is a positive154 eigenvector of A for the eigenvalue reB)

l/r(B) which is such that for all A E a( A), I AI > r(~). 15 5

For simplicity, now consider the mono-kinetic case (G = 1). We can then, as in Example 3, define e E H b(Q) such that de = 1. On applying (with the notations of Example 3) the 1st Krein-Rutman theorem to B, E., and P., we deduce that the smallest eigenvalue of A is simple156 (with, as always, a positive eigenvector). The study of the case which involves many groups (and notably of the case with two groups) will be developed in Chap. XII, §6.

154 That is to say, a positive function. 155 Note that A has a pure point spectrum and that a(A) = {l/A, A E a(B), A of. O}. 156 Note that A is here self-adjoint in the complexified space H, and thus has a real spectrum with inferior bound.

Documents

Mathematical Analysis and Numerical Methods for Science and Technology || Spectral Theory