REFERENCEIC/66/62

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

7- • *

r - <

EIGENFUNCTION EXPANSIONSASSOCIATED WITH THE SECOND ORDER

INVARIANT OPERATORON HYPERBOLOIDS AND CONES

N. LIMlCj . NIEDERLE

AND

R. R^CZKA

, •/

1966PIAZZA OBERDAN

TRIESTE

IC/66/62

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

EIGENFUNCTION EXPANSIONS ASSOCIATED WITH THE SECOND ORDER

INVARIANT OPERATOR ON HYPERBOLOIDS AND CONES r

N. LIMIC*

J. NIEDERLE**

and

R. RA.CZKA***

TRIESTE

June 1966

t To be submitted to the Journal of Mathematical Physics

* On leave of absence from Institute Rudjer BoskoviC, Zagreb.* * On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague.

***On leave of absence from Institute of Nuclear Research, Warsaw.

ABSTRACT

The eigenf unction expansions associated with the second order in-

variant operator on hyperboloids and cones are derived. The global unitary

irreducible representations of the SO» (p, q) groups related to hyperboloids

and cones are obtained. The decomposition of the quasi-regular represent-

ations into the irreducible ones is given and the connection with the Mautner

theorem and nuclear spectral theory is discussed.

-1 -

EIGENFUNCTION EXPANSIONS ASSOCIATED WITH THE SECOND ORDER

INVARIANT OPERATOR ON HYPERBOLOIDS AND CONES

" 1. INTRODUCTIONi

1) 2)In our two previous works the most degenerate irreducible in-

finitesimal representations of an arbitrary non-compact rotation group3)4)SO(, (p, q) have been derived. These representations have been related

to the homogeneous spaces SO0(p, q)/SOp(p-l,q) , SO0(p, q)/SO0(p,q-1) and

SO0(p, q)/T p*<1** ® SO(p-l,q-l) which can be represented by the hyper-

boloids H^ , H? and by the cone C respectively. These homogeneousi| spaces are of rank one under the action of the group. The infinitesimal

!: irepresentations have been constructed by means of the sets of harmonic

functions associated with the second order invariant operators related to

the above-mentioned manifolds,

J The completeness of these sets of harmonic functions has been proved^ 5) 6)

in the present work by using the classical TITCHMARSH '-. KODAIRA ;

I eigenfunction expansion theory associated with an ordinary second order

differential equation. Moreover, it is shown that the carrier spaces of theF

irreducible infinitesimal representations of the SO0 (p, q) group are also theI i

carrier spaces (after completion) of the global unitary irreducible represent-

ations of the group SO0(p, q).

• 1 •

In Sec. 2 the review of the main "results in the form convenient for\ applications is given. The other sections are devoted to the proofs. Thusi1 Sec. 3 contains the proof of the essential self-adjointness of the Laplace-Beltrami operator on the linear; manifold jjfX) , which was introduced

2) 'earl ier . The proof of the completeness of the harmonic functions which

1)2) !

were constructed in our previous papers is given in Sec. 4. In Sec. 5

we prove the unitarity and irreducibility of the representations of the group

5Q>(ftf.) anc* consider the decomposition of the quasi-regular represent-

ations into irreducible ones. In Sec. 6 we show an interesting connection

of our approach to the eigenfunction expansion and the general theory of7)the eigenfunction expansion developed by GEL'FAND and KOSTIUCHENKO ,

MAURIN and GARDING . Appendix I contains some auxiliary comput-

- 2 -

ations. In Appendix n we review the main results on the representations

of the compact group SO(p) ,

2. REVIEW OP MOST DEGENERATE REPRESENTATIONS OF SOfl (p, q)

GROUPS

We have considered in our previous papers three homogeneous

spaces X of rank one under the action of the non-compact rotation group

SOe(ftf) (see (2. 2) and (4.1) '). They can be represented by the hyper-

boloids H£ and H« and the cone C f :/. r • - ?*

f 1 for the hyperboloid Hf, p^ q

rx')V ic*nl- fan-• • • - f t ^ A < ° f o r t h e c o n e c z '(, -1 for the hyperbolbid Hy, p> q .

(2.1)

These homogeneous spaces are imbedded in the (p + q)-dimensional

Minkowski space Mri/r . Thefconsidered homogeneous spaces are para-

metrized by use of biharmonic .co-ordinates £2. which are defined in Sec. 32)

for the hyperboloids and in Sec/. 4 for the cone.

Let cfcCfl-} be the Riemannian left-invariant measure on X and let

be the Hilbert space of L j/<-)-type.

ation of the rotation group is determined by

be the Hilbert space of L j/<-)-type. The quasi-regular represent-

(2.2)

The quasi-regular representation defined by (2. 2) is unitary but reducible.

Its irreducible parts are given below for any particular choice of the homo-

geneous space X . Since the homogeneous spaces H^ , Hf and cf areof rank one under the action of the 50o (fij,) group, the ring of the invariant

operators of the representation of the corresponding Lie algebra

is generated only by one operator.

The generators X?y of the Lie algebra Jt^)are represented by

the unbounded operators Ah' in the Hilbert space %(x) (see (6. 2) ,1) 2) 2)1) 2) 2)

(6. 3) , (4. 4) and (4. 5) ' ) . Hence if we consider the representations ofthe generators and their polynomials (for instance the Casimir operator)

- 3 -

we must restrict their domain of definition to some dense linear manifold

in JcfX) . The most convenient choice is the common invariant domain

Q)(X) for all the operators X,j' , i.e., $fc) has the property Xy £)&<) CC £$() for any X»' .

Let Y)(X) be the linear manifold determined by vectors y€ JcdXj.dL

the form ^{h)= "Pfa4..-- tX^ *" jGtP\—'21 C*lJ // where r ( V " / ^ / i sa polynomial in X . . . . . X * and the X;, f * ^ ?/ - * v / " ^ are expressed in

1) 2)biharmonic co-ordinates -C2 as in Sec. 3 and Sec. 4 . Then the linear

manifold ^ / y is a dense invariant domain of the operators Xu' .

The proof can be found in Appendix I.

Let us review now the particular cases in detail:A) Hyperboloid H | , p $ q > 2

i I

! i) The spectrum Sf^C^JJ of the Laplace-Beltrami operator

(see (3.10) ) consists of ^he discrete spectrum '1 ~L(l +ft*± '2. ) , L = - (?—%_ -} -p- lAl / ] • +1, . . . and the continuous

spectrum CS(j\(Mf )):

1) 2)ii) The corresponding eigenf unctions are (see Sec. 3 r and Sec. 2 )

where

^

. . ' • .

The functions ' ^tr.-fm^{^) are defined in Appendix p in the ex-

pressions (A.1) - (A..6). The same expressions hold for the functions

/ - ) but in tilde-variables and tilde-indices.

The variable A is independent of the indices /,• ,#*,• /,• and/*%-

whereas L has the following dependence:

- ^ - 2 " / **0,1,1 r . . . (2.3)

iii) The* completeness relations:

>/

kn * * *»f * c * ^ /

where the left-invariant measure <&*(&) on H* is defined by

and (/fcfuJ) is defined in Appendix II by (A. 9).

- 5 -

where tM^ is the set of values of indices Yz/--- , /nitH] * which are

restricted by the conditions (A. 4), (A. 5) and (2. 3) and J(\ the set

restricted by the conditions (A. 4) and (A. 5).

iv) Fourier transform:

where <5" denotes either the variable L or A and 2. is the

range of <T corresponding to the whole spectrum of /^Hgjt

v) The complete classification of the irreducible representations

can be ,made by means of the spectrum of the operator Z\(H^) and

the spectrum of the operator ~P , where ' is defined in (5. 4)

(On the subspace *3t C 'Jt(Sj corresponding to a point of ~PS(A(H^)

the operator r has a unique eigenvalue (~)} ''whereas on the linear

subset 2r corresponding to a point of CSCAiM£)j; f has

the "eigenvectors" with both its eigenvalues). We denote by f a

pair (ffp where p is the eigenvalue of ~P : p~ ^ i . Let <^(v*

be that subset of tAtf for which either / ^ ] +-^£%} * s even and

p = 1 or ${%\ + ^t&} ^ s ot*^ a n t * P = "1 • (The set cA , is already

of this type, while c/r* has two proper subsets of this type.) We

consider the unitary space Js of -e -type determined by sequences

where

(Let us keep in mind that the scalar product in Jy is determined by

The completion of the unitary space ju with respect to the norm

' //• • \\t is the Hilbert space %^ .

•i The unitary irreducible representation of the group SO{p, q) and the

irreducible representation of the algebra j\(i)f &) are defined by

! '• - 6 - ' ' ' • '

/

*>-+4 [J x

it

where ^ / is defined by (2. 2) and ) ( ) • / by (6. 2)1 ', (6. 3)H (4.4)2)

and (4. 5)2).

vi) The quasi-regtilar representation in (2. 2) decomposes into the

irreducible representations in the following way

in the Hilbert space (see Sees. 4 and 5)

B) Hyperboloid

Changing q ^± p in all the expressions of A) and removing the

"tilde" from any variably or index which previously had it and at

the same time placing a "tilde" over any variable or index which

previously did not have it, we obtain the corresponding expressionsor

related to the hyperboloid H b .

C) Hyperboloid H£ , p > 2' i i i < ^ " • •

Replacing the function YJ'f ..'f JXt%f%) by the function ^ . &P (tM*% )

and the index / ^ by \\m4 \ (as ^ = \A«4 / ) and putting q = 2 in A)

we obtain the desired expressions related to the hyperboloid H £ .

- 7 -

completeness relations have the form

D) Hyperboloid Up , p > 2

First we change q ^± p in all the expressions of A) and remove the

"tilde" from any variable or index which previously had it and at the

same time place a "tilde1! over any variable or index which previously

did not have it .and put a - 2. Then we r^niarp V~" ' Z? J% )

and c L - ' - ^ otherwise.

-9-3

where c/V is the set of values of indices o£f /lr •y'ir>cfii] restricted

by (A. 4), (A. 5), (2.4) and (2. 5) and <Jfa the set of values of indices

^ ^ ^ restricte

measure dk(&) is defined by

< ,,( / ,n restricted by (A. 4) and (A. 5). The left invariant

iv) The Fourier transform:

••X

v) The unitary irreducible representations are classfied by use of

both the spectra of the operators A>\n4) and / . (See Sec. 3 )

We denote by £ the pair edf p and by cn^ that subset of the set

for which either oi+ -cf/vi is even and p = 1 or o(. + -££%} is

odd and p = - 1 . We consider the unitary space *U determined by

the sequences

The unitary irreducible representation of the group 50e (ft i) and the

irreducible representation of the corresponding Lie algebra are

defined on z/ in an analogous way as in A).

on the Hilbert space / ^ . ,=2*%' * fay' ®fa % ~faF) Hyperboloid Hp , P > 1. (Upper sheet of

The Laplace-Beltrami operator A(Hp) has only a continuousspectrum 5/A/fHp')) .'. tf i (£d' f~ Af[0,^) • T h e cor-responding eigenfunctions are

The completeness relations and the Fourier transform have the same

structure as in A). The unitary irreducible represepatationspf

$Q>(p 4.) related to the/upper sheet of the hyperboloid Up are

classified by Af A$Loif°) • The unitary space X) is determin-

ed by the sequences

The unitary irreducible representations of the group and the irreduc-

ible representation of the algebra are defined as in A). The decom-

position of the quasi-regular representation is:

(/.•> fee/A0* MO •* if

G) Cone C ^ . (Upper sheet of C\ )

i) The second order invariant Q^ (4- 2) n a s on^v t n e continuous

spectrum S^J- A^ /

ii) The corresponding eigenfunctions are

where

iii) The completeness relations have the form

where the left invariant measure on C £ is

iv) The Fourier transform is defined by

v) The unitary irreducible representations are classified by the

spectrum of jf Q — (£'£~,2jz and r . Let us denote by f theQ (

pair A.p, A $(-**,*») p€ \.-^i)) an(* le* u s consider the unitary

space yj determined by the sequences

y > , ^ tlM

where c/ir is the subset of the set cn^ restricted by the condition

that either is even and p = l or

odd and p = - 1 . The unitary irreducible representation of the group

and the irreducible representation of the algebra are defined as in.A),

vi) The decomposition of the quasi-regular representation into the

irreducible ones has the form+ exo -f-eO

ur A-

on + *o

-12-

3. ESSENTIAL SELF-ADJOINTNESS OF AfX) ON ©DC)

To simplify the notations we denote by / ; /»*, < and '/>*> the sets

respectively. The set of values of the indices £2r-, fipnt'*'iff/*tfzt'/,l <£

^{PAi "*'l "I ***1%1 ' ** e " ' t h e s e t °* i n t e 6 e r s restricted by the conditions

(A-4), (AS), we denote by cM •

The restriction of the differential operator AfXy which was derived

in refs.l)2) to the dense invariant linear manifold SfyX) is denoted by A.

The operator ^ is symmetric on J0P</• We shall prove in this sectionthat ^X is the essentially self-adjoint operator on OCX) for any particular

case of the homogeneous space X.

A) Hyperboloid Hgj, p > q > 1

The operator

' 2 . :

symmetHc operator on Jzx)ysatisfying /&,£& ~ //»,fa on

J a dense linear manifold in eOAJ •• ^ A T has the unique extension

to the projector T(>m$£ on <JtiiA) , the range of which is the closed sub-

space Xf^fcfX) . The subspaces <%!&,/£• fX) with a different set of

indices are orthogonal to each other and their sum over all - /*v ^/** £ «/»

is equal to the space ^c (Xy . This follows from the density of &\/J ina n d t n e completeness of the orthogonal set of harmonic functions

i n t h e Hilbert space ^i(S^') (For the completeness of the set of

see Appendix tt )

every subspace &&*,?£&) reduces A to the symmetric operator

-13-

-15-

Let us choose as %&) and Ut(&) those solutions of

for which

One can easily verify by direct calculation the possibility of this choice.i

Then by rough estimate of the third term in (3. 6) we find that this term

vanishes faster than rfr at the origin and its derivative faster than

, so that the first two terms describe the behaviour of 0(&) at the origin.1 The domain jyfA J of the closure A is the linear manifold determined by

< those ft §)f$)i or which 3(-/,^)~0 . 4 &£)(%*). where 3(f,s) = &'**(#&)>'\ $'(4) - gtyffy)) *3> As the domain 20) of the closure A* of the

1 symmetric operator A on X)(A J contains JoWyit has to be BC^C, #J ~ 0

for /c:?=('fcUd%?(de)e></>(-2eJ,l&) , *e (0,«>)l ffr). / }f £)CA').| This condition implies b = 0 in (3. 6). Then any two vectors $> % € 2)(A J

\ satisfy ~B(a a ) - 0 , i. e., the adjoint A of ^ is defined on

j pM"4 A i . e . , ^ is self-adjoint on <J6Y/l / . In other words, A is

] essentially self-ad joint on ,£)(ASJ.

The analogous proof holds for q - Z, -^ = 0,i

From Proposition 3.1 and the decomposition Jc(X/—

5 it follows:

Proposition 3. 2. The operator &• is essentially self-adjoint on jOCX),

i The domain 0fa Jot the self-ad joint operator *~± ~ ^ is determined by

! vectors fe %fX) for which 1A, ?z f f 0(D%;/#) and XThe operator Z/ on z)fA / is defined by ^^Z = X"

B) Other hyperboloids H^ and H^ .

As the form {3. 2) of the operator A holds for any type of hyper-

boloid, the proof of the essential self-adjointness of ZH on bOiX) can be

carried out in an analogous way to A).

-16-

pC) Cone C^ , p > q > 1.

The operator A in the present case is defined by

J, The:associated differential equation

O. ( 3 . 7 )

has the solutions %£(r)= e*P {(' i * $±^f~X )&r } / AgainAS is the self-adjoint on S)CAS)B.S for non-real \ one of i\k(r) , t'-^Z

has the property ^ (•)$<£(<>, c) for every c > 0 ; whereas the other has

not this property and the same relation holds at infinity.

4. EIGENFUNCTION EXPANSION

1)2)In two previous works we used the set of those eigenfunctions

j £, (JLj with the eigenvalue X of the differential operator /^Oy or «£

which were restricted by the boundary condition / ,*,£ <JW — t' where

. Let us prove now that this set constitutes the complete set of

ns of

previous section.

eigenf unctions of the self-adjoint operator A which was defined in the

P <•

A) Hyperboloid Hg , p ^ q > 1.so.

We have shown that the operator A is reduced to the operator

J)S^ ?~ o n every subspace %„ fa (X) . Then by the mapping V the

operator U^f~ has been mapped to the operator A (3. 2) and the

space < % ^ W o n to the space «(. (o,»°) . In this way the problem of theso.

eigenfunction expansions associated with the operator Zs is reduced to

the investigation of the eigenfunction expansions associated with the operator

A or, in other words, associated with the second order ordinary dif-

ferential equation (3.4). Except for Z{fy\ f f - 2 . and ^{%i=0/%=3 there

is the unique solution of the eigenfunction expansion associated with the4)differential equation (3.4) and this expansion is just that one which is

-17-

associated with the self-adjoint operator A . In two exceptional cases

we have to impose boundary conditions in such a way that the corresponding

expansions are associated with the self-adjoint operator A . For

h%i}~°t %= z/ t h e b e n a v i o u r o f t h e function fjf{&)f g€ 0(^Caj^ is des-cribed by (3. 6) where b = 0 as proved. As lim ft*" ~Bfa,Q)=o according

to the proof of Proposition 3.1 f we conclude that ' j^(\}~ to and con-

sequently only the function /Ui(^-) enters the expression for the eigen-4)

function expansion. A similar conclusion holds for the case /(%y&, 7=3 i

only that eigenfunction f (&) of {3.4) enters the expression for the

eigenfunction expansions for which f (0) •* 0.i

The differential equation (3.4) has the form of the Schrodingeri ' 15)

differential equation and we prefer to use the method of JOST and KOHN3)4)adopted for such equations rather than the general method ' '. The solution

of the differential equation (3. 4) which is regular at the origin has the form

(4.1)

where C = €r%. i + M and

The solutions of (3.4) which behave like exp^ -f^l /C^^J -A / are

These two solutions are generally singular at the origin. The connection

among the solutions (4.1) and (4. 3) can be easily described by introducing

the variable/* ^ - ^ ^ + u(t^Z2f- j , and denoting

-18-

1

KytiH+lfyt) (4.4)

where

±fe*£*+M) rfC)r(±fE*£3+jk))1 . (4. 5)

Let us introduce now the function

( 4 - 6 )

(The function G0uf4($') is analytic in the half-plane Re/4 > -

having poles at zeros of the function YVJ.£*) , i . e . , at the points

The number of poles is finite and we choose R in the contour C of Pig. 1

large enough to enclose all the poles. Let us multiply the equality

ffa , fa £ Res QP**> W')by ^f$') and integrate over the interval (ot cx>) , where TC^J is

infinite differentiable function vanishing faster than any power in # for

large t?' . The limiting value of the the expression obtained as 71-* txt

has the form:

OQ( 4 > 7 )

Here the function Wd Qt) f jU * - &£? + f(£!£*f^ X is essentially

the Titchmarsh spectral function An(X) which determines the spectrum

-19-

Proposition 4 .1 . The mappings

7

are, one to the other, inverse isometric mappings: F F-J w %(X) and

FF" = I (m %/S) • The symbol s.lim in (4.12) means the strong

convergence £-*>/ /h e 0(y<) and 5. lim in (4.13) means the strong

convergence * - * *y r , : ^ /"< Vi^,jJ>; f.e&fr), *£ SC^V, /,**,,\ T n e operator A a i s diagonally (or spectrally) represented in

5. UNITARITY, IRREDUCIBILITY AND MAUTNER'S DECOMPOSITION

The unitary representation SO6(p;p$ f -^ UeZ • = (Z*J% (tyf h*M Sf^), 4s»,fafc4^l 6 J^S) is reducible. (This representation

and the quasi-regular representation (2. 2) are unitarily equivalent with

respect to the operator r of Proposition 4.1). The carrier space of an

irreducible representation is necessarily an eigenspace 3P ' with an

eigenvalue A € Sf-A ) . If the eigenspace 'X ' corresponds to the

proper eigenvalue, i. e. , to \( P5 y , then 3£ ' is a proper subspace

of the Hilbert space 7P(S) . (The image of these spaces under F are

the irreducible spaces considered in ref. 1)). In the case of the re-

presentation of the non-compact group the invariant operator may have a

continuous spectrum besides a discrete one as in our case. The "eigen-

space" jE corresponding \O%€CS(A> / i s not a proper subspace of the

Hilbert space l£[S) and one needs to extend such eigenspace to a certain

-22-

Hilbert space in order to obtain a faithful representation on a Hilbert space.

Then one expects that the Hilbert space 3i(Sj will be decomposed into the

sum over J?A ; X f R5 (ASA) and an integral of # \ A ^ £S &*).

Such decomposition of the Hilbert space %(S) into a so-called direct17)

integral of Hilbert spaces was investigated by von NEUMANN .

Let us try to construct the desired decomposition of the Hilbert

space 26(S) . The structure of the Hilbert space %($) leads us to con-

sider a unitary space of £— type determined by vectors X^'~ / X^

), ',*, fa t^l} „ where Z^K^/Z-X A*x ^Q) X X(X) —* ZsJirfr) i s n o t bounded mapping we shall

restrict ourselves to the mapping ( \J> *

and justify this by the following two propositions:

Proposition 5.1. For every '/,/**, •£,/£ € (J& , /f 2>0^) the function

is analytic in the A -plane cut along the segment (-JFlJL})•

and tends to zero faster than any polynomial in % along the positive real

axis. Moreover for every bounded subset 3 C ^A^Sf^f-t ^ ^ A / there

exists a /€ &CK) such that XA/J (/)?<>** B, ]

Proof: Let /_> be any bounded convex domain in the cut A -plane.

Using expressions (4.1) and (4. 3) and the definition of the mapping

of Sec. 3 we easily find the uniform bound of the functions V - (•&) in

the domain ZJ X X *

fa Yiiwhere the constants A(£J and B(S) depend on the domain ZJ . The

analyticity of XXJ,%C/) in the cut A-plane follows from *he analyticity

of the function ~y^ £ (SI) in the cut A -plane for any SL*X , the bound

(5.1) and the bound l/^)j< Cc/,% e*p (-Zcfy ) , /£ 96$?) Theasymptotic behaviour of 7i ^ ^ (/•) along the positive real axis follows

from the equality A* X^C/h/rA*i(*$A*$r](Sl)^JL),^ theinvariance of 0O<) under ^ and the uniform bound / Y*£% WIKe*p(M#) on £XX , where R , =, {" A y {^}f-^ x ± „

The second part of the proposition follows from its first part and the

- 2 3 -

density of jDfX) in %6<).

Using Proposition 5.1 the following is easily proved:

Proposition 5. 2. The linear manifold rt) determined by vectors

XX : = {x*£$s&) ; t,**,?^ <*JG, , /*$&)} is dense in the Hilbert^space «$? of / -type, vectors of which are sequences Xx- = { %• *-•»}

j ~ . £.&ltfx\ for which // %H\ = 27 / XX5\l < •*•* r , l <*

Let (yi — ^t,^.,* be an orthonormal complete sequence in the

Hilbert space $(X) which is obtained by the Smith orthogonalization

process from a sequence $ % . . . determining 0{A) . In our case

the denumerable set of vector fields $fe?V * ~* XX.' = [xX^(=fh)j

ffA"t C/>* ( ^C f ^ * ^ j forms a so—called fundamental family of

vector fields. * 9 J 2 O ) The vector field Sfe?V * A - * • • V* •' •=i s c a U e d A'measurable if / X.A, f * )^ =

is a >(,-measurable function for all n . The

direct integral

of the Hilbert spaces 7c is the Hilbert space & of equivalent classes

of the JA.-measurable vector fields XX for which J ftXxl% e(fit$ < **

and where the scalar product is defined by

( 5 ' 3 )

The Hilbert spaces ^uX/ and Ol[5) are isometrically isomorphic

with respect to the Fourier transform F ,and %[S) is represented as the

direct integral of Hilbert spaces 7c ,

Unitarity. We obtained the Hilbert spaces ** , closed subspacesrJt ' of which may be the carrier spaces of the unitary irreducible

representations of the group SQCfig) . We shall first prove the unitarity

of the representations on *" for every j^f S(Zi y . (This induces

the unitarity of the representation in any of % >ut ). For the proof of the

unitarity we need the auxiliary;

- 2 4 -

Proposition 5. 3. The properties stated in Proposition 5,1 for the functionC/); y € Q(X) are also valid for the function %

Proof: From the definition of the function X ^^ (Ufrf) a n d

invariance of the measure Cwi^-) it follows

( 6 -4 )

Now the proof of Proposition 5.1 may be applied as T ar (ffSLJ has

essentially the same bound as in the proof of Proposition 5.1 because the

finite point SL. under the action of Q£ SOQ (fig.) is transformed again

to a finite point J 2 / = 9 - ^

Proposition 5.4. The representation

is norm preserving in oc/ and has unique unitary extension Vq in

£)* for every X€ Sfe?*)-

Proof: The unitarity of the representations of the group S0o (h%) o n

'TP^ X€ PSC^1) w a s considered in ref. 1) In fact,there we worked with

the spaces F^jc )• N o w w e &ve t n e general proof valid also for X £ CSC& j ./ I Asct_

Because of the strong commutativity of U* and Zi for anyu -measurable function F(^) , uniformly bounded on 5 [A J w e have

- (/, F(*?*)h )*o/- JAcgXk). . (5.6)

According to Proposition 4.1 we can rewrite (5. 6) in the form:

Because of the continuity of ((£x) Ljf X)x - (x.) f X) on CS(^(Propositions 5.1 and 5. 3) and boundedness on P5(A, ) and because of the

arbi t rar iness of the uniformly bounded function r(^) on S ( A y w e

-25-

conclude (C$x) t$f*)x - (x>f\ , k*Sfr*),x)tX€ £>X . As 3*is the dense linear manifold in ^ t Up can be uniquely extended to the

unitary operator Va in X = 3) . (The Hilbert spaces % are

representatives of <& and V» is the representative of ys of the

decomposable operator IM =Jsfep) ty ^ ^ ) m % SsdfV ^ &

The following proposition is not directly connected with our consider-ations but it shows a deeper property of the linear manifold Su .

Proposition 5. 5. The operators Xf7 are essentially skew-adjoint on 3)\

Proof: The "opera tors X,V are skew-symmetric operators on oO as

Vr*&) is unitary in ^ A and / / ^ T ^ ~ V - X~ ]xX II^ -> 0 for

%X£ J&*1 a s ^ "* ° • L e t u s f i r s t s n o w t n a t a n y v e c t o r X^t £) is ananalytic vector for the operators Lw ( L •• is the representation of the

/} 1 ) *

generator tif of the one-parameter compact subgroup ). First oneproves that —^ Z;'* is an invariant operator using the commutation rules.Hence it is proportional to —AfS^'J — A(S^~ J as the representations arethe most degenerate ones. We do not lose on generality, supposing that theconstant of the proportionality is equal to one. For any %* € $"* the de-

composition X ' - Z * } ^ , X ^ fa e ^

are finite-dimensional vector spaces determined by vectors A ^

/ ^ 2){X) w i t h f i x e d ^iB& i 1%) 6 ^X »b a s a f i n i t e number of non-vanish-ing terms. We easily find the bound // £.j %h \\\ < p ^ UAtlAt fp-2) +

(£».**+2-*)] ^ A " A from the expression

Now using the irreducibility of the algebra determined by Ljj in the sub-

spaces «$V /"-, we find the estimate

( 5 - 8 )

From the inequality (5. 8) we easily see that XA£ & is an analytic vector

-26-

of Lj-j and consequently I;: is essentially skew-adjoint on Si .

The estimate for Hfl B:J X*k ,. XXi 0 * c a n be obtained in an<*=f * / * • - . " f ••• ••• . ; • ' : • • • ' - ' • ' • • : ; . ' V •• • ; ' . ; ' £ ' ' '

analogous way starting with the invariant operator ^\&ir)~~ Z ( ^ v on «z/;

n(5.9)

It follows again that '%£'%/ is an analytte vector for O7)' and that p//*i . ' • ' . •• •• . i / , •' ~ • - • •'.

is essentially skew-adjoint on ^ . '

Irreducibility. Let us try to find the invariant subspaees *«£•• of the

space % with respect to the representation of the group (5* 8).

1)2)In our previous work we have considered the representations of

the Lie algebra J\C/°,^) of the group §Qo(fi%} on the Lie algebra of the

unbounded operators X,)' in fthe Hilbert space J£ >* 6 Sf/fy . The: : • ! / • • • • : • ' . ••, • • • . • • • • • . . . ; ' . / : ' y A ' . .

irreducibility with respect to the algebra of the operators ^>j has beendefined in the following way: The representation 3t(f>tf)z X;y

reducible on a common invariant domain oZ/ if for any two %rf

an operator A exists in the enveloping algebra such that (XAf A f1

The irreducible representation of the Lie algebra were found in refs. 1)2)

and they are reviewed in Sec* 2 of the present work. It was established there

that the common invariant domain 0 ' has the following structure:

lx'"-Z •e'vOT,%.X

where ^A*,* £rz,i a r e finite-dimensional vector spaces determined by

vectors X l^C^J t j?GXs(X) with fixed -ffl%} , */&} €WA. and

is an infinite subset of the set tn^ • ;

Proposition 5. 6. The representation (5. 5) is irreducible on the closed1 i

-27-

subspace J: ' = <ZJ ' of the space

Proof: Every finite-dimensional subspace <* 7 is irreducible

with respect to the representation Lj>' of the Lie algebra \<y' of the

maximal compact subgroup 5>OCf>) YSOPp. This fact and the continuity

of the operators l/Y in the Hilbert spaces ^ i « ^ f t i induces the

irreducibility of the representation SO(/>)XS0($)->g -> Uo X f i

as, for instance, A Uj ~{JjA~O in . S ^ i ^ . Induces Alj'~ Lj'A •= O

in ^ V . £/•£? • Hence any invariant subspace ^ , /r=--jj 2 , - . . - of the

space J ^ / W with respec t to the represen ta t ion of the group •SOo(f>l<£) must

have the form %£ =21 © 5^.A r where t/V is an infinite subset

of the set cAjn . Then we can introduce the operator /< , eigenspaces

of which are ^ , with the eigenvalue //£ . From Kl)g -{J&K' = O in

#Jt a n d t h e f a c t t h a t ^ ' ^ is the common invariant domain

of fyjffRfr- we conclude tfty ~ BjK^O in .0A '^ which contradictsthe results of refs. 1) and 2). f

Mautner1 s decomposition. We have determined the unitary irreduc-

ible representations of the group S0c (p,%) related to three homogeneous

spaces H^ , H^ and C& . Let us now discuss the consequences which

follow from the comparison of our results with the general theory. Let

J/?v be the smallest weakly closed *"-algebra generated by the set of

operators (2. 3) Ua t 4€ G related to the homogeneous space X and let

be the commutant of y^x • W e know that the centre

5rv " J^x^ ^x i s S e n e r a t e d by t h e operators • F(Asa) >

where Ffj\) = (\ + i/& T . (The operator A ^ i s unbounded so that we

have to consider its function F(A. J which is a bounded operator in

The decomposition of the Hilbert space %(S) into the direct integral (5. 2)

is so-called central decomposition. We found that the Hilbert spaces tJt

in (5. 2) are generally reducible so that the central decomposition does not

lead generally to decomposition into the irreducible representations. We

have to add a certain number of discrete operators in order to classify the

irreducible representations. (Let us recall that the ring of the invariant

- 28 -

operators in the enveloping algebra is generated by Z\ ), According to22)

MA-UTNER's theorem this means that the maximal commuting sub-

algebra 07/^ of the commutant JC% is generally generated by more

operators than Ffe?*) . In particular cases we have the following

situations:

a) For SOCfiz),p>Z>Zt

[ FfA*)f P } and #?C£ m- £>£*J, P}; where the parity

operator 7 is defined by (5^4) ;

b) For S6(f>,%) f>^<£^2. X WHZ~{Ffa*V /P,T}) where

operator / is defined in point v) of D) jof ,Sec* 2, ^uP **•• \

f

As the spectra of the operators I and / have only a finite number

of points, it is easy to find the Mautner decomposition of the, Hilbert space

%($) with respect to the maximal commutative algebra W?x . The

decomposition for any particular case is written in Sec* 2.

In ref. 1) we have given the maximal set of commuting operators A

(7. 8) for every irreducible representation. These operators are un-

bounded and we have to replace them by the bounded operators F (A)y

Vs) . The operators A of the set (7.8) are in the enveloping

algebra and according to the preceding discussion we have iri general to

complete the operators F(A ) with a certain number of discrete operators

in order to generate the maximal commutative algebra on JciX),

6. NUCLEAR SPECTRAL THEORY :

The existence of the complete set of generalized eigenf unctions for a

self-adjoint operator on a separable Hilbert space Ais was first establish-7)

ed by GEL'FAND and KOSTIUCHENKO '. They introduced a nuclear space

, such that the Hilbert space *C is the completion of <p with

respect to one of the norms in <» and the dual <p of continuous function-

als and they considered the triplet <p CitOfc which we shall call the

- 2 9 - i

GEL'FAND-KOSTIUCHENKO20) triplet. Their theory can be put in a

very elegant form using Maurin's construction of the nuclear space <p for

an arbitrary^ strongly commutative denumerable set of normal operators/ 8)19) _ x . . ,. . . . . . 19)23)20)

A A . Let us state now the nuclear spectral theorem \ and then

let us show how the results derived in the present work may be a nice

illustration of this general theory.Nuclear spectral theorem:

1°. Such a nuclear space <p exists that 'p^- Tc^^fi is a Gel'fand-

Kostiuchenko triplet •

2°. Each A A maps <& continuously into <& j

3°. There exist a direct integral JL ~L 1% cfdO] and such a Hilbert

isomorphism r ! /£~*> % that for-'(T-almost all. A jA Aand H are common generalized eigenspaces of a ll A-

A ^£X y>y identically in ¥>6^ , where

* j^ $ jy~L is the spectrum of Aw . Taking in each Ji

the orthonormal basis Qf} n~ -4f z, • •-, «w**f we obtain the

generalized Fourier-Plancherel equation

(9; f) - / Z

4°. The isomorphism F is defined by

5°. The spectral synthesis of ¥ r g y p j

The nuclear space <+? of the spectral theorem can be obtained using8)19)23)

MAURIN's construction. According to this construction as a nuclear

space <p i for the operator zl (or the maximal commutative algebra

of jx% , may be taken to be the linear manifold *£)()(), which has been

exploited so far very extensively, furnished, of course, with certain nuclear

topology (for details s

continuous functionals

topology (for details see ref. 19)). The dual space £) is the space of the

-30-

It was proved in Sec. 4 that the set of functionals (6.1) constitute a

complete set of eigenfunctionals. The direct integral, the Fourier trans-

form F and the spectral synthesis were explicitly written in Sec. 4 if only

the operator A** is considered. It is easy to generalize the expressions

of Sec. 4 for the maximal commutative algebra.

ACKNOWLEDGMENTS

The authors would like to thank Professors Abdus Salam and P. Budini

and the IAEA for hospitality kindly extended to them at the International

Centre for Theoretical Physics, Trieste.i

The authors are grateful to Professor K. Maurin not only for much

valuable insight and many private communications but also for placing at

their disposal the manuscript of his forthcoming book. They are grateful

to Dr. 3. Fischer for some remarks and Dr. S. Woronowicz for interesting

discussions.

-31-

APPENDIX I

The proof of the density of SDOy in Jc(X) . Let us suppose that

some 9e ZfCX), 9 7^0} exist, such that (^tflrO for every J~£ &C>().

We introduce the local co-ordinates -y* k-tZi^^fi+f.-) in the Minkowski

space M?'^ round the hyperboloids Hg and Up by <%*= oX* f>0)

where X are biharmonic co-ordinates and round the cone Co by H -

We define the function hfy,#/= e*p (2fz<i4h - 2cA*i9) for f^fj-/£) and

zero otherwise. The function v/VU7.. ••,/y/>^=(^Jf?,^.) has the proper1-

H

As the set of functions ^ V - >y y P**) - Pfff2..-,where ify'H1... p^h^ a polynomial, determine a dense linear manifold

, Such a function ilfaW... -w^x i s t s that (-U^lfO. But

this contradicts our assumption as ( ' ^ y^ -= J^v(f)/C//L(Q.) hfo

€?(2fAG) cf(Jl) O

The proof of the invariance of SO(Xj with respect to X « ' . The in-

variance follows from the equality

An analogous equality holds for S$%j . Similarly* the skew-symmetry of

of the operator Yu' can be proved and consequently the symmetry of

3)0Q.on

-32-

APPENDIX II

Review of most degenerate representations of SO(p) groups.

i) The Laplace-Beltrami operator

has only a discrete spectrum 5 {Ws^~'J) ' -

defined in (A. 6) ', (A. 7) '

* P~ hp<i\ " °> 1> li''V

ii) The corresponding eigenfunctions in our biharmonic system arer 7

* e 2 ' k

'

where^ is the set of variables *lj* . , ^ , *^. . , F :.Vkf fa£), V** L°,2irhand the indices in (A*) are ( ' j . ^ / ^ / j . ^ ^ *

(A.2)

-3)

Here

ditions

"*/

4*4-

The normalization factors in (A.I) are

restricted by the con-

i,

iii) The completeness relations:25)

:

-33-

- ( A - 6 )

- 7 )

f (A 8)

where J\ is the set of values of indices ^2r--j miP/i] restricted by (A. 4)

and (A. 5), and dk(w) is the left invariant measure on S defined byr

(A; 9)• 2i-j

iv) There is no need to go to the space of Fourier transforms to

define the unitary irreducible representations. Therefore, we consider

the space itCSr ) , The unitary irreducible representations are class-

ified by the number ^fi/A , ^(P/\ - 4 */ -2,— and they are defined by

(u, rt::,where %? ^(S^~r) is the subspace of the space T^fe*"*) determined by

the vectors > ,*, ,., />»rJ?) with fixed

v) The decomposition of the quasi-regular representation (2. 2) into

the irreducible ones is trivial as <%:(5 J = Z- & *>• (Sr A

-34-

REFERENCES AND FOOTNOTES

1) R. RACZKA, N. LIMIC and J. NIEDERLE, Discrete degenerate

representations of non-compact rotation groups, ICTPf Trieste,

preprint IC/66/2. (To be published in J. Math. Phys.).

2) N. LIMIC, J. NIEDERLE and R. RACZKA, Continuous degenerate

representations of non-compact rotation groups, ICTP, Trieste,

preprint IC/66/18. (To be published in J. Math. Phys.).

3) The representations of the compact SO(p) group related to the sphere

and the Lorentz-type group SO(p, 1) related to the real Lobachevski

space {Up in our notation) are given in the book by N. J. VILENKIN: ~

Special functions and theory of group representation, ''NAUKA",

Moscow (1965) (In Russian).

4) For example,the representations of some lower-dimensional SO(p, q)

groups are given in:

J.A.G. ALCARAS,, and P. L. FERREIRA, J. Math. Phys. .6, 578 (1965);

V. BARGMANN, Ann. Math. 48, 568 (1947);

C.G. BOLLINI and J. J. GIAMBIAGI, Unified representations of the

rotation and Lorentz (2 + 1) groups, preprint, University of Buenos

Aires, 1965;

J. DIXMIER, Bull. Soc. Math. France 89, 9(1961);

A.Z. DOLGINOV, Soviet Phys. -JETP 3. 589 (1956); '

A.Z. DOLGINOV and J. N. TOPTYGIN, Soviet Phys. -JETP 10, 1022

(1960);

A.Z. DOLGINOV and A. N. MOSKALEV, Soviet Phys. -JETP 10, 1202

(1960);

J.B. EHRMAN, Proc. Cambridge Phil. Soc. 53, 290 (1956);

N. T. EVANS, Discrete series for the universal covering group of

the 3 + 2 de Sitter group, preprint, University of Cambridge 1965;

A. KIHLBERG, Arkiv Fysik 27, 373 (1964), 28, 121 (1964) and

30, 121 (1965);

A. KIHLBERG and S. STROM, Arkiv Fysik (to appear);

A. KIHLBERG, Chalmers University preprint.

A. KIHLBERG, On the unitary representations of a class of pseudo-

orthogonal groups, preprint, Gothenburg, 1965;

-35-

L.H. THOMAS, Ann. Math. 42, 113 (1941).

5) E. C. TITCHMARSH, Eigenfunction expansions, P a r t i , Clarendon

Pre s s , Oxford 1962.

6) K. KODAIRA, Am. J. Math. 71, 921 (1949).

7) I. M. GEL'FAND and A. G. KOSTIUCHENKO, Dokl. Akad. Nauk SSSR103, 349 (1955).

8) K. MAURIN, Bull. A cad. Polonaise Sci. 7, 471 (1959).

9) L. GARDING, Seminar on Applied Mathematics, Boulder, Colorado,

pp. 1-30,(1957),

10) In fact we mean the spectrum of the self-adjoint extension of

(For details see Sees. 3 and 4).

11) We use notation: [%1's %. if a = 2r and (a-f)/z ifa=2r+ilr-~ff?,...j

i&=2r and (A+fl/l if a =

12) The cf-function cT^-il//tjis defined by: f<$(Jl-S$n)J(il')o(k(Sl')=

13) M.H. STONE, Linear transformations in HUbert space, Am. Math.

Soc. CoUoq. Pub. , Vol XV, New York (1964), (Chapter X).

14) The lemma 5. 6 of ref. 3).

15) R. JOSTandW. KOHN, Phys. Rev. 87, 977 (1952).

16) A.C. ZAANEN, An introduction to the theory of integration, North

Holland Pub. Co. - Amsterdam (1961).

17) J. von NEUMANN, Ann. Math. 50, 401 (1949).

18) M.A. LAVRENTIEV and B. V. SHABAT, Methods of theory of

functions of complex variables, "NAUKA", Moscow (1965) (In Russian).

- 36 -

W.

19) K. MAURIN, The methods of Hilbert space, MIR, Moscow (1965)

(In Russian).

20) K. MAURIN, General eigenfunction expansions and group represent-

ations (Lecture Notes), ICTP, Trieste, preprint IC/66/12.

21) E. NELSON, Ann. Math. 70, 572 (1959).

22) F.L MAUTNER, Ann. Math. 51, 1 (1950).

23) K. MAURIN, Manuscript of the forthcoming book: Unitary Represent-

ations of Non-compact Groups and Associated Eigenfunction Expansions.

24) We use d-function as defined in:

M. E. ROSE, Elementary Theory of Angular Momentum (John Wiley

1961) p. 53.

25) The proof of the completeness of the set of harmbnic functions

fa) i n J£(SrV in another co-ordinate system js given

in ref. 5).

-37-

Tie//

- 38 -

t

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