InfStepQPerScat2

8/2/2019 InfStepQPerScat2

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SCATTERING THEORY FOR SCHRODINGER OPERATORS ON

STEPLIKE, ALMOST PERIODIC INFINITE-GAP

BACKGROUNDS

KATRIN GRUNERT

Abstract. We develop direct scattering theory for onedimensional Schrodinger

operators with steplike potentials, which are asymptotically close to differentBohr almost periodic infinitegap potentials on different halfaxes.

1. Introduction

One of the main tools for solving various Cauchy problems, since the seminalwork of Gardner, Green, Kruskal, and Miura [4] in 1967, is the inverse scatteringtransform and therefore, since then, a large number of articles has been devoted todirect and inverse scattering theory.

In much detail the case where the initial condition is asymptotically close top(x) = 0, has been studied (see e.g Marchenko [ 13]). Taking this as a startingpoint, there are two natural cases, which have also been considered in the past. Onthe one hand the case of equal quasi-periodic, finite-gap potentials p(x) = p+(x)and on the other hand the case of steplike constant asymptotics p(x) = c withc = c+. Very recently, the combination of these two cases, namely the case thatthe initial condition is asymptotically close to steplike, quasi-periodic, finite-gap

potentials p(x) = p+(x), has been investigated by Boutet de Monvel, Egorova,and Teschl [1].

Of much interest is also the case of asymptotically periodic solutions, which hasbeen first considered by Firsova [3]. In the present work we propose a completeinvestigation of the direct scattering theory for Bohr almost periodic infinite-gapbackgrounds, which belong to the socalled Levitan class. It should be noticed,that this class, as a special case, includes the set of smooth, periodic infinitegapoperators.

To set the stage, we need:

Hypothesis H.1.1. Let

0 E0 < E1 < < En < . . .be two increasing sequences of points on the real axis which satisfy the following

conditions:

(i) for a certainl > 1,

n=1(E2n1)

l(E2n E2n1) < and(ii) E2n+1 E2n1 > Cn

, where C and are some fixed, positive

constants.

2000 Mathematics Subject Classification. Primary 34L25; Secondary 34L40.

Key words and phrases. Scattering theory, Schrodinger operator.Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.

1


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2 K. GRUNERT

We will call, in what follows, the intervals (E2j1, E2j) for j = 1, 2, . . . gaps. In

each closed gap [E2j1, E2j] , j = 1, 2, . . . , we choose a point

j and an arbitrary

sign j {1, 1}.Next consider the system of differential equations for the functions j (x),

j (x),

j = 1, 2,..., which is an infinite analogue of the well-known Dubrovin equations,given by

dj (x)

dx= 2j (x)

(j (x) E0 )

j (x) E2j1

j (x) E2j(1.1)

k=1,k=j

j (x) E2k1

j (x) E2k

j (x) k (x)

with initial conditions j (0) = j and

j (0) =

j , j = 1, 2, . . .

1. Levitan [9],

[10], and [11], proved, that this system of differential equations is uniquely solvable,

that the solutions j (x), j = 1, 2, . . . are continuously differentiable and satisfyj (x) [E2j1, E2j ] for all x R. Moreover, these functions j (x), j = 1, 2, . . .are Bohr almost periodic 2. Using the trace formula (see for example [9])

(1.2) p(x) = E0 +

j=1

(E2j1 + E2j 2j (x)),

we see that also p(x) are real Bohr almost periodic. The operators

(1.3) L := d2

dx2+p(x), dom(L) = H2(R),

in L2(R), are then called an almost periodic infinite-gap Schrodinger operator ofthe Levitan class. The spectra ofL are purely absolutely continuous and of theform

= [E0 , E

1 ] [E2j , E2j+1] . . . ,

and have spectral properties analogous to the quasi-periodic finite-gap Schrodingeroperator. In particular, they are completely defined by the series

j=1(

j ,

j ),

which we call the Dirichlet divisor. These divisors are associated to Riemann sur-faces of infinite genus, which are connected with the functions Y

1/2 (z), where

(1.4) Y(z) = (z E0 )j=1

(z E2j1)E2j1

(z E2j)E2j1

,

and where the branch cuts are taken along the spectrum. It is known, that theSchrodinger equations

(1.5)

d2

dx2 +p(x)

y(x) = zy(x)with any continuous, bounded potential p(x) have two Weyl solutions (z, x)and (z, x), which satisfy

(z, .) L2(R), resp. (z, .) L2(R),

1We will use the standard branch cut of the square root in the domain C\R+ with Im

z > 0.2 For informations about almost periodic functions we refer to [12].


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SCATTERING THEORY FOR SCHRODINGER OPERATORS 3

for z C\ and which are normalized by (z, 0) = (z, 0) = 1. In ourcase of Bohr almost periodic potentials of the Levitan class, these solutions havecomplementary properties similar to the properties of the Baker-Akhiezer functionsin the finite-gap case. We will briefly discuss them in the next section.

The object of interest, for us, is the one-dimensional Schrodinger operator L inL2(R)

(1.6) L := d2

dx2+ q(x), dom(L) = H2(R),

with the real potential q(x) C(R) satisfying the following condition

(1.7)

0

(1 + |x|2)|q(x) p(x)|dx < ,

for which we will characterize the corresponding scattering data with the help ofthe transformation operator, which has been investigated in [6].

2. The Weyl solutions of the background operators

In this section we want to summarize some facts for the background Schr odingeroperators L of Levitan class. We present these results, obtained in [6], [9], [15],and [16], in a form, similar to the finite-gap case used in [1] and [5].

Let L be the quasi-periodic one-dimensional Schrodinger operators associatedwith the potentials p(x). Let s(z, x), c(z, x) be sine- and cosine-type solutionsof the corresponding equation

(2.1)

d

2

dx2+p(x)

y(x) = zy(x), z C,

associated with the initial conditions

s

(z, 0) = c

(z, 0) = 0, c

(z, 0) = s

(z, 0) = 1,

where prime denotes the derivative with respect to x. Then c(z, x), c(z, x),s(z, x), and s(z, x) are holomorphic with respect to z C\. Moreover, theycan be represented in the following form

c(z, x) = cos(

zx) +

x0

sin(

z(x y))z

p(y)c(z, y)dy,

s(z, x) =sin(

zx)

z+

x0

sin(

z(x y))z

p(y)s(z, y)dy

The background Weyl solutions are given by

(z, x) = c(z, x) + m(z, 0)s(z, x),

resp. (z, x) = c(z, x) + m(z, 0)s(z, x),

where

m(z, x) =H(z, x) Y1/2 (z)

G(z, x), m(z, x) =

H(z, x) Y1/2 (z)G(z, x)

,

are the Weyl function of L (cf [9]), where Y(z) are defined by (1.4),

(2.4) G(z, x) =j=1

z j (x)E2j1

, and H(z, x) =1

2

d

dxG(z, x).


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4 K. GRUNERT

Using (1.1) and (2.4), we have

(2.5) H(z, x) =1

2

d

dx G(z, x) = G(z, x)

j=1

j (x)Y1/2

(j (x))

ddzG(

j (x), x)(z j (x)) .

The Weyl functions m(z, x) and m(z, x) are Bohr almost periodic.

Lemma 2.1. The background Weyl solutions, for z C, can be represented in thefollowing form

(z, x) = expx

0

m(z, y)dy

=

G(z, x)G(z, 0)

1/2exp

x

0

Y1/2 (z)

G(z, y)dy

,

(2.6)

and

(z, x) = exp

x0 m(z, y)dy

=G(z, x)

G(z, 0)1/2

exp

x

0

Y1/2

(z)

G(z, y) dy

.

If for some > 0, |z j (x)| > for allj N andx R, then the following holds:For any 1 > > 0 there exists an R > 0 such that

|(z, x)| e(1)x Im(z)

1 +DR|z|

, for any |z| R, x > 0,

and

|(z, x)| e(1)x Im(z)

1 +DR|z|

, for any |z| R, x < 0,

where D denotes some constant dependent on R.

As the spectra

consist of infinitely many bands, let us cut the complex plane

along the spectrum and denote the upper and lower sides of the cuts by uand l. The corresponding points on these cuts will be denoted by

u and l,respectively. In particular, this means

f(u) := lim0

f( + i), f(l) := lim0

f( i), .

Defining

(2.7) g() = G(, 0)2Y

1/2 ()

,

where the branch of the square root is chosen in such a way that

(2.8)1

i

g(u) = Im(g(u)) > 0 for

,

it follows from Lemma 2.1 that

(2.9) W((z), (z)) = m(z) m(z) = g(z)1,where W(f, g)(x) = f(x)g(x) f(x)g(x) denotes the usual Wronskian determi-nant.

For every Dirichlet eigenvalue j = j (0), the Weyl functions m(z) and m(z)

might have poles. If j is in the interior of its gap, precisely one Weyl function


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m or m will have a simple pole. Otherwise, ifj sits at an edge, both will havea square root singularity. Hence we divide the set of poles accordingly:

M = {j | j (E2j1, E2j) and m has a simple pole},M = {j | j (E2j1, E2j) and m has a simple pole},M = {j | j {E2j1, E2j}},

and we set Mr, = M M M.In particular, we obtain the following properties of the Weyl solutions (see, e.g.

[2], [9], [18], and [17]):

Lemma 2.2. The Weyl solutions have the following properties:

(i) The function (z, x) (resp. (z, x)) is holomorphic as a function of z inthe domainC \ ( M) (resp. C\( M), real valued on the setR\ ,and have simple poles at the points of the setM (resp. M). Moreover, they

are continuous up to the boundary u l except at the points fromM and(2.10) (u) = (l) = (l), .

For E M the Weyl solutions satisfy

(z, x) = O

1z E

, (z, x) = O

1

z E

, as z E M,

where the O((z E)1/2)-term is independent ofx.The same applies to (z, x) and

(z, x).

(ii) At the edges of the spectrum these functions possess the properties

(z, x) (z, x) = O(

z E) near E \M,

and(z, x) + (z, x) = O(1) near E M.

(iii) The functions (z, x) and (z, x) form an orthonormal basis on the spec-trum with respect to the weight

(2.11) d(z) =1

2ig(z)dz,

and any f(x) L2(R) can be expressed through

(2.12) f(x) =

R

f(y)(z, y)dy

(z, x)d(z).

Here we use the notation

f(z)d(z) :=u

f(z)d(z) l

f(z)d(z).

3. The direct scattering problem

Consider the Schrodinger equation

(3.1)

d

2

dx2+ q(x)

y(x) = zy(x), z C,


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6 K. GRUNERT

with a potential q(x) satisfying the following condition

(3.2)

0

(1 + x2)|q(x)

p

(x)|dx 0 for int((2)). Thus, using thecorresponding properties of (z, x), finishes the first part.

(ii) By (3.19) the reflection coefficient can be represented in the following form:

(3.30) R() = W((), ())W((), ())

= W((), ())W()

,

and is therefore continuous on both sides of the set int()\(M M).Moreover,

|R()| =W((), ())W()

,where the denominator does not vanish, by assumption and hence R()is continuous on both sides of the spectrum in a small neighborhood of theband edges under consideration.

Next, let E {

E2j

1, E2j

}with W(E)

= 0. Then, if E

M

, we canwrite

R() = 1 j,()W(() (), ())W()

,

which implies R() 1, since () () 0 by Lemma 3.1 as E. Thus we proved the first case.If E M with W(E) = 0, we use (3.30) in the form

R() = 1 j,()W(() + (), ())W()

,

which yields R() 1, since j,() 0 and () + () = O(1) byLemma 3.1 as

E. This settles the second case.

4. The Gelfand-Levitan-Marchenko Equation

The aim of this section is to derive the Gelfand-Levitan-Marchenko (GLM)equation, which is also called the inverse scattering problem equation and to obtainsome additional properties of the scattering data, as a consequence of the GLMequation.


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14 K. GRUNERT

-

-

-

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.......

.........

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.......

....................................................................................

......

............ ........ ....... ....... .......... ............. ........... ........ .......

.......

.......

............

.........

......

.................................

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......................

...................

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.......................

.............. .............. ........ ........... ....................... ................... ................ ............ ......... ........ ........ ................................................................................................................................................................

................ .......... ........... ........ ....... ......... .............................................................................

Figure 1. Contours ,n

Therefore consider the function

G(z, x, y) = T(z)(z, x)(z, y)g(z) (z, x)(z, y)g(z):= G(z, x, y) + G

(z, x, y), y > x,

where x and y are considered as fixed parameters. As a function ofz it is meromor-phic in the domain C\ with simple poles at the points k of the discrete spectrum.It is continuous up to the boundary u l, except for the points of the set, whichconsists of the band edges of the background spectra + and , where

(4.1) G(z, x, y) = O((z E)1/2) as E + .

Outside a small neighborhood of the gaps of + and , the following asymp-totics as z are valid:

(z, x) = eizx(1+O( 1

z))

1 + O(z1/2)

, g(z) =1

2i

z+ O(z1),

(z, x) = eizx(1+O( 1

z))

1 + O(z1)

, T(z) = 1 + O(z1/2),

(z, y) = eizy(1+O( 1

z))

1 + O(z1)

,

and the leading term of (z, x) and (z, x) are equal, thus

(4.2) G(z, x, y) = eiz(yx)(1+O( 1

z))O(z1), y > x.

Consider the following sequence of contours ,n,, where ,n, consists of two

parts for every n N

and 0:(i) C,n, consists of a part of a circle which is centered at the origin and has

as radii the distance from the origin to the midpoint of the largest band of[E2n, E

2n+1], which lies inside

(2), together with a part wrapping aroundthe corresponding band of at a small distance, which is at most , asindicated by figure 1.

(ii) Each band of the spectrum , which is fully contained in C,n,, is sur-rounded by a small loop at a small distance from not bigger than .


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W.l.o.g. we can assume that all the contours are non-intersecting.Using the Cauchy theorem, we obtain

12i

,n,

G(z, x, y)dz =

kint(,n,)Resk

G(z, x, y), > 0.

By (4.1) the limit value of G(z, x, y) as 0 is integrable on , and the functionG(z, x, y) has no poles at the points of the discrete spectrum, thus we arrive at

(4.3)1

2i

0,n,

G(z, x, y)dz =

kint(0,n,)Resk

G(z, x, y), y > x.

Estimate (4.2) allows us now to apply Jordans lemma, when letting n , andwe therefore arrive, up to that point only formally, at

(4.4)1

2i G(,x,y)d = kd

Resk

G(,x,y), y > x.

Next, note that the function G(,x,y) does not contribute to the left part of

(4.4), since G(u, x , y) = G(

l, x , y) for (1) and, hence

(1)

G(,x,y)d =

0. In addition,

G(,x,y)d = 0 for x = y by Lemma 2.2 (iv).Therefore we arrive at the following equation,

(4.5)1

2i

G(,x,y)d =kd

Resk

G(,x,y), y > x.

For making our argument rigorous, we have to apply Jordans lemma, whichimplies that the contribution of the integral along the circle of C0,n,, convergesagainst zero as n and we have to show that the series of integrals along theparts of the spectrum contained in C0,n, converges as n

. This will be done

next.Using (2.12), (3.3), (3.7), (3.18), and Lemma 2.2 (iv) we obtain

1

2i

G(,x,y)d =

T()(, x)(, y)d()

=

R()(, x) + (, x)

(, y)d()

=

R()(, x)(, y)d() +

(, x)(, y)d()

x

dtK(x, t)

R()(, t)(, y)d() + (t y)

= Fr,(x, y)

xK(x, t)Fr,(t, y)dt + K(x, y),

where

Fr,(x, y) =

R()(, x)(, y)d().

Now properties (ii) and (iii) from Lemma 3.2 imply that

|R()| < 1 for int((2)), |R()| = 1 for (1) .


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16 K. GRUNERT

and by (2.6) we can write

Fr,

(x, y) =

R

()

(, x)

(, y)d

()

=

R()(G(, x)G(, y))1/2

2Y()1/2exp((, x) + (, y))d,

with

(, x) := x

0

Y()1/2

G(, )d iR.

We will show

Lemma 4.1. The sequence of functions

Fr,n,(x, y) =0,n,

R()(, x)(, y)d(),

is uniformly bounded with respect to x andy, that means for alln N, |Fr,n,(x, y)| C. Moreover, Fr,n,(x, y) converges uniformly as n to the function

Fr,(x, y) =n=0

(E2n,E

2n+1)

R()(, x)(, y)d(),

which is again uniformly bounded with respect to x and y. In particular, Fr,(x, y)is continuous with respect to x and y.

Proof. For as we have the following asymptotic behavior(i) in a small neighborhood Vn of E = E

n

|R

()

(, x)

(, y)g

()|

= OE2j E2j1

( E) ,(ii) in a small neighborhood Wn of E = E

n , if E

R()(, x)(, y)g() = exp(i

(x + y)(1 + O(1

))O(

1

),

(iii) and for \jN(V

n Wn )

(4.6)

R()(, x)(, y)g() = exp(i

(x + y)(1 + O(1

)))C

+ O

13/2

.

These estimates are good enough to show that Fr,(x, y) exists, if we choose Vnand Wn in the following way: We choose V

n (1) (2), if En is a band edge

of (1)

, such that V

nconsists of the corresponding band of

(1)

together with the

following part of (2) with length E

n En1, if n is even and En+1 En , if n

is odd. IfE+n is a band edge of (2), we choose Vn (2), where the length of

Vn is equal to the length of the gap pf next to it. We set Wn (2) with

length 3(En En1), if n is even and 3(En+1 En ), if n is odd, centered at themidpoint of the corresponding gap in . As we are working in the Levitan classand we therefore know that

n=1(E

2n1)

l(E2n E2n1) < for some l > 1,we obtain that the sequences belonging to Vn and W

n converge.


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For the last sequence, observe first that

|exp(

i

(x + y)O(1

))

| (x + y)O(

1

), if (x + y)O(

1

)

/2,

respectively

| exp(i

(x + y)O(1

))| 1 + (x + y)O( 1

), if (x + y)O(

1

) /2.

Furthermore for |x + y| 1,ba

exp(i

(x + y))C

d = exp(i

(x + y))

2C

i1/2(x + y)|ba

ba

exp(i

(x + y))C

i3/2(x + y)d,

and for |x + y| 1,

b

a

exp(i(x + y))C

d = exp(i(x + y)) 2Ci1/2 |ba

ba

exp(i

(x + y))C(x + y 1)

d

+

ba

exp(i

(x + y))C

i3/2 d,

which yieldsba

exp(i

(x + y))C

d = exp(i

(x + y))

2C

i1/2(2 x y) |ba

+ b

a

exp(

i

(x + y))C

i3/2

(2 x y)d.

Similarly we haveba

exp(i

(x + y))(x + y)O(1

3/2)d = exp(i

(x + y))O(

1

)|ba

+

ba

exp(i

(x + y))O(1

2)d

For showing the convergence of the corresponding series, we can use the followingargument: Integrate (4.6), where C can be computed explicitly, from E0 to andsubtract the parts corresponding to the neighborhoods of the gaps, which can beestimated using

|E2j1,E2j exp(i

(x + y))C

d

| C

E2j1(E2j

E2j

1),

and the fact that Vn and Wn are situated around the gaps and have length smaller

than 3 times the length of the corresponding gap. Thus, putting all together, alsothe last part of the integral is finite as we are working in the Levitan class and thebound is independent of x.

For investigating the other terms, we will need the following lemma, which istaken from [3]:


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18 K. GRUNERT

Lemma 4.2. Suppose in an integral equation of the form

(4.7) f

(x, y)

x

K

(x, t)f

(t, y)dt = g

(x, y),

y >

x,

the kernel K(x, y) and the function g(x, y) are continuous for y > x,|K(x, y)| C(x)Q(x + y),

and for g(x, y) one of the following estimates hold

(4.8) |g(x, y)| C(x)Q(x + y), or

(4.9) |g(x, y)| C(x)(1 + max(0, x)).Furthermore assume that

0

(1 + |x|2)|q(x) p(x)|dx < .

Then (4.7) is uniquely solvable for f(x, y). The solution f(x, y) is also contin-uous in the half-plane y > x, and for it the estimate (4.8) respectively (4.9) isreproduced.

Moreover, if a sequence gn,(x, y) satisfies (4.8) or (4.9) uniformly with respectto n and pointwise gn,(x, y) 0, for y > x, then the same is true for thecorresponding sequence of solutions fn,(x, y) of (4.7).

Proof. For a proof we refer to [3, Lemma 6.3].

Remark 4.3. An immediate consequence of this lemma is the following. If |g(x, y)| C(x), where C(x) denotes a bounded function, then |g(x, y)| C(x)(1 +max(0, x)) and therefore |f(x, y)| C(x)(1 + max(0, x)). Rewriting this in-tegral equation as follows

f(x, y) = g(x, y) x

K(x, t)f(t, y)dy,

we obtain that the absolute value of the right hand side is smaller than a boundedfunction C(x) by using (1.7) and (3.4), and hence the same is true for the lefthand side. In particular if C(x) is a decreasing function the same will be true forC(x).

We will now continue the investigation of our integral equation.


Fh,n,(x, y) =

(1),u 0,n,

|T()|2(, x)(, y)d()

is uniformly bounded, that means for all n N, |Fh,n,(x, y)| C(x), whereC(x) are monotonically decrasing functions as x . Moerover, Fh,n,(x, y)converges uniformly as n to the function

Fh,(x, y) =

(1),u

|T()|2(, x)(, y)d(),

which is again bounded by some monotonically increasing function. In particular,Fh,(x, y) is continuous with respect to x and y.


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Proof. On the set (1) both the numerator and the denominator of the function

G(,x,y) have poles (resp. square root singularities) at the points of the set

(1)

(M ((1)

+ (1)

)) (resp. (1)

(M\(M M)) , but multiplyingthem, if necessary away, we can avoid singularities. Hence, w.l.o.g., we can suppose

(1) (Mr,+ Mr,) = . Thus we can write

1

2i

(1)

G(,x,y)d =1

2i

(1)

T()(, x)(, y)g()d.

For investigating this integral we will consider, using (3.18),

1

2i

(1)

T()(, x)(, y)g(, y)d

=1

2i

(1)

(, x)

(, y) + R()(, y)

g()d.

First of all note that the integrand, because of the representation on the right hand

side, can only have square root singularities at the boundary (1) and we thereforehave

(1) [E2n1,E2n]

|(, x)

(, y) + R()(, y)

g()|d

2

(1) [E2n1,E2n]

|(, x)(, y)g()|d

C(y)C(x) (E2n E2n1)

E0+

E2n E2n1

E0

,

where E2n1 and E2n denote the edges of the gap of in which the corresponding

part of (1)

lies and C(x) denote monotonically decreasing functions from nowon. Therefore as we are working in the Levitan class and by separating

(1) into

the different parts, one obtains that

| 12i

(1)

T()(, x)(, y)g()d| C(y)C(x).

Thus we can now apply Lemma 4.2, and hence

| 12i

(1)

T()(, x)(, y)g()d| C(y)C(x)(1 + max(0, y)).

Note that we especially have, because of (3.4),

| 12i (1) T()(, x)(, y)g()d| C(x)

Therefore we can conclude that for fixed x and y the left hands side of (4.5) existsand satisfies

| 12i

(1)

T()(, x)(, y)g()d| C(x),

and hence

| 12i

G(z, x, y)dz| C(x).


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20 K. GRUNERT

Furthermore, since (, x) R as (1) , we have

12i

(1)

G(,x,y)d = 12i

(1),u

(, y)

(, x)W()

(, x)W()

d

Moreover, (3.18) and Lemma 3.2 (ii) imply

(, x) = T()(, x) T()T()

.

Therefore,

(, x)W()

(, x)W()

= (, x) 1

W()+

T()

T()W()

T()(, x)

W()(4.10)

= (, x) 2Re(T1

()W())T()|W()|2 T()()

W().

But by (3.22)

T1 ()W() = |W()|2g() iR, for (1) ,

and therefore the first summand of (4.10) vanishes. Using now W = (Tg)1 wearrive at

(, x)

W() (, x)

W()= |T()|2g()(, x)

and hence

1

2i

(1)

G(,x,y)d = Fh,(x, y) x

K(x, t)Fh,(t.y)dt,

where

Fh,(x, y) =

(1),u

|T()|2(, x)(, y)d(),

and

|Fh,(x, y)| C(x)C(y)

by Lemma 4.2. The partial sums Fh,n,(x, y) can be investigated similarly

We will now investigate the r.h.s. of (4.3) and (4.5). Therefore we consider firstthe question of the existence of the right hand side:

To prove the boundedness of the corresponding series on the left hand side, itis left to investigate the series, which correspond to the circles. We will derivethe necessary estimates only for the part of the nth circle KRn, , where Rn,denotes the radius, in the upper half plane as the part in the lower half plane can


21/24


be considered similarly. We have

| KRn, G(z, x, y)dz|

0 Ce

R(x

y)(1

) sin(/2)

d

/2

0

CeR(xy)(1) sin()d

/2

0

CeR(xy)(1)2

d

C 1R(x y)(1 ) e

R(xy)(1)2

|20 ,

where C and denote some constant, which are dependent on the radius (cf.Lemma 2.1). Therefore as already mentioned the part belonging to the circlesconverges against zero and hence the same is true for the corresponding series.

Thus we obtain that the sequence of partial sums on the right hand side of (4.3)and (4.5) is uniformly bounded and we are therefore able to proof the followingresult:


Fd,n,(x, y) =

kd0,n,(k )

2(k, x)(k, y)

is uniformly bounded, that means for all n N, |Fd,n,(x, y)| C(x), whereC(x) are monotonically increasing functions. Moreover, Fd,n,(x, y) convergesuniformly as n to the function

Fd,(x, y) = kd

(k )2(k, x)(k, y),

which is again bounded by some monotonically increasing function. In particular,Fd,(x, y) is continuous with respect to x and y.

Proof. Applying (3.3), (3.14), (3.15), (3.25), and (3.27)to the right hand side of(4.5), yields

kd

Resk

G(,x,y) = kd

Resk

(, x)(, y)

W()

= kd

(k, x)(k, y)

W(k)ck,

= kd

(k )2

(k, x)(k, y)

= Fd,(x, y) x

K(x, t)Fd,(t, y)dt,

where

Fd,(x, y) :=kd

(k )2(k, x)(k, y).


22/24

22 K. GRUNERT

Thus we obtained the following integral equation,

Fd,(x, y) = K(x, y) Fc,(x, y)

x K(x, t)Fc,(t, y)dt(4.11)

x

K(x, t)Fd,(t, y)dt,

which we can now solve for Fd,(x, y) using again Lemma 4.2 and hence Fd,(x, y)exists and satisfies the given estimates. The corresponding partial sums can beinvestigated analogously using the considerations from above.

Thus we have proved the following theorem

Theorem 4.6. The GLM equation has the form

(4.12) K

(x, y) + F

(x, y)

x

K

(x, t)F

(t, y)dt = 0,

(y

x) > 0,

where

F(x, y) =

R()(, x)(, y)d()(4.13)

+

(1),u

|T()|2(, x)(, y)d()

+

k=1

(k )2(k, x)(, y).

Moreover, we have

Lemma 4.7. The function F

(x, y) is continuously differentiable with respect toboth variables and there exists a real-valued function q(x), x R with

a

(1 + x2)|q(x)|dx < , for all a R,

such that

(4.14) |F(x, y)| C(x)Q(x + y),

(4.15)

ddx F(x, y) C(x)

q

x + y

2

+ Q(x + y)

,

(4.16)

a

ddx F(x, x) (1 + x2)dx < ,where

Q(x) = x2

|q(t)|dt,

and C(x) > 0 is a continuous function, which decreases monotonically as x .


23/24


Proof. Applying once more Lemma 4.2, one obtains (4.14). Now, for simplicity, wewill restrict our considerations to the + case and omit + whenever possible. SetQ

1(u) =

uQ(t)dt. Then, using (3.2), the functions Q(x) and Q

1(x) satisfy

a

Q1(t)dt < ,a

Q(t)(1 + |t|)dt < .

Differentiating (4.12) with respect to x and y yields

|Fx(x, y)| |Kx(x, y)| + |K(x, x)F(x, y)| +x

|Kx(x, t)F(t, y)|dt,

Fy(x, y) + Ky(x, y) +

x

K(x, t)Fy(t, y)dt = 0.

We already know that the functions Q(x), Q1(x), C(x), and C(x) are monotonicallydecreasing and positive. Moreover,

x

q+x + t2 + Q(x + t)Q(t + y)dt (Q(2x) + Q1(2x))Q(x + y),thus we can estimate Fx(x, y) and Fy(x, y) can be estimates using (3.5) and themethod of successive approximation. It is left to prove (4.16). Therefore consider(4.12) for x = y and differentiate it with respect to x:

dF(x, x)

dx+

dK(x, x)

dxK(x, x)F(x, x)+

x

(Kx(x,tF(t, x)+ K(x, t)Fy(t, x))dt = 0.

Next (3.4) and (4.14) imply

|K(x, y)F(x, x)| C(a)C(a)Q2(2x), for x > a,where

a (1 + x

2)Q2(2x)dx < . Moreover, by (3.5) and (4.15)

|Kx(x, t)F(t, x)|+ K(x, t)Fy(t, x) 4C(a)C(a)qx + t2 Q(x + t) + Q2(x + t),together with the estimates

a

dx x2x

Q2(x + t)dt a

|x|Q(2x)dx supxa

x

|x + t|Q(x + t)dt < ,a

x2x

qx + t2

Q(x + t)dt a

Q(2x)dx supxa

x

qx + t2

(1 + (x + t)2)dt < ,and (3.6), we arrive at (4.16).

In summary, we have obtained the following necessary conditions for the scat-tering data:

Theorem 4.8. The scattering data

S=

R+(), T+(), u,l+ ; R(), T(), u,l ;

1, , 2, R \ (+ ), 1 , 2 , R+

(4.17)


24/24

24 K. GRUNERT

possess the properties listed in Theorem 3.2, 3.3, 3.4, and3.5, and Lemma4.1, 4.4,and 4.5. The functions F(x, y) defined in (4.13), possess the properties listed inLemma 4.7.

Acknowledgements. I want to thank Ira Egorova and Gerald Teschl for manydiscussions on this topic.

References

[1] A. Boutet de Monvel, I. Egorova, and G. Teschl, Inverse scattering theory for one-dimensionalSchrodinger operators with steplike finite-gap potentials, J. dAnalyse Math. 106:1, 271316,

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[3] N.E. Firsova, The direct and inverse scattering problems for the one-dimensional perturbedHill operator, Math. USSR Sbornik 58, no.2, (1987).

[4] C.S. Gardner, J.M. Green, M.D. Kruskal, and R.M. Miuram, Method for solving theKortewegde Vries equation, Phys. Rev. Lett. 19, 10951097, (1967).

[5] F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions, Volume1: (1+1)-dimensional continuous models, Cambridge studies in advanced mathematics 79,Cambrige University Press, Cambridge, 2003.

[6] K. Grunert, The Transformation Operator for Schrodinger Operators on Almost PeriodicInfinite-Gap Backgrounds, J.Diff.Eq. (to appear).

[7] H. Kruger, On perturbations of quasi-periodic Schrodinger operators, J.Diff.Eq. (to appear).[8] H. Kruger and G. Teschl, Effective Prufer angles and relative oscillation criteria, J. Diff. Eq.

245, 38233848 (2008).

[9] B.M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, 1987.[10] B.M. Levitan, Approximation of infinite-zone by finite-zone potentials, Math. USSR-Izv, 1,

5587, (1983).[11] B.M. Levitan, Approximation of infinite-zone potentials by finite-zone potentials, Math.

USSR-Izv 20, no. 1, 5587, (1983).[12] B.M. Levitan and V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cam-

bridge University Press, Cambridge, 1982.

[13] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkhauser, Basel, 1986.[14] F.S. RofeBeketov and A.M. Kohlkin, Spectral Analysis of Differential Operators; Interplay

Between Spectral and Oscillatory Properties, World Scientific Monograph Series in Mathe-matics 7, World Scientific, Singapore, 2005.

[15] M. Sodin and P. Yuditskii, Almost periodic SturmLiouville operators with Cantor homoge-

neous spectrum, Comment. Math. Helv. 70, no.4, 639658, (1995).[16] M. Sodin and P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum,

infinitedimensional Jacobi inversion, and Hardy spaces of characterautomorphic functions,J. Geom. Anal. 7, no.3, 387435, (1997).

[17] G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schrodinger

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[18] E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equa-tions, Clarendon Press, Oxford, 1946.

Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria

E-mail address: [email protected]: http://www.mat.univie.ac.at/~grunert/
mailto:[email protected]://www.mat.univie.ac.at/~grunert/http://www.mat.univie.ac.at/~grunert/http://www.mat.univie.ac.at/~grunert/http://www.mat.univie.ac.at/~grunert/mailto:[email protected]

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