On An inverse spectral problem for Sturm-Liouville operator with discontinuous coefficient Текст научной статьи по специальности «Математика»

ISSN 2074-1863 Уфимский математический журнал. Том 7. № 3 (2015). С. 125-137.

УДК 517.984

ON AN INVERSE SPECTRAL PROBLEM

FOR STURM-LIOUVILLE OPERATOR WITH DISCONTINUOUS COEFFICIENT

KH. R. MAMEDOV, D. KARAHAN

Abstract. In this paper, the direct and inverse problems for Sturm-Liouville operator with discontinuous coefficient are studied. The spectral properties of the Sturm-Liouville problem with discontinuous coefficient such as the orthogonality of its eigenfunctions and simplicity of its eigenvalues are investigated. Asymptotic formulas for eigenvalues and eigenfunctions of this problem are examined. The resolvent operator is constructed and the expansion formula with respect to eigenfunctions is obtained. It is shown that eigenfunctions of this problem are in the form of a complete system. The Weyl solution and Weyl function are defined. Uniqueness theorems for the solution of the inverse problem according to Weyl function and spectral date are proved.

Keywords: Sturm-Liouville operator, expansion formula, inverse problem, Weyl function. Mathematics Subject Classification: 34A55, 34B24, 47E05

1. Introduction

In non-homogeneous environment, vibration, diffusion and other physical problems are described by differential equations with discontinuous coefficient [1]-[8]. While solving these problems, the spectral properties of Sturm - Liouville problem with discontinuous coefficient should be analyzed [7]-[21].

We consider the differential equation

-y" + q(x)y = A2p(x)y, 0 ^ x ^ n (1.1)

with the boundary conditions

y'(0) = 0, y(n) = 0 (1.2)

where A is a complex spectral parameter. We suppose that q E L2(0, n) is a real-valued function, p is a piecewise continuous function:

,,il0 ^ x < a , s

p(x) = < 2 . . (1.3)

J \ a2 a ^ x ^ n. v '

As p(x) = 1, the same problems were studied in [4], [22]-[24]. In general case, problem (1.1), (1.2) is solved by considering it in the intervals [0,a) and [a,n]. But in this study, problem (1.1), (1.2) was reduced to an equivalent integral equation in [0, n] interval. For the special solution of problem, the integral representation in this interval was used. There were shown the simplicity and reality of eigenvalues and the orthogonality of the eigenfunctions associated with different eigenvalues and the asymptotic formulas for the eigenvalues and eigenfunctions were obtained.

Kh. R. Mamedov, D. Karahan, On an inverse spectral problem for Sturm-Liouville operator with discontinuous coefficient.

© Kh. R. Mamedov, D. Karahan 2015. Поступила 22 апреля 2015 г.

The definition of the system of eigenfunctions is given in the weighted space L2,p(0, n) with the weight p. The expansion formula for eigenfunctions and Parseval identity were obtained. In the last part of the study, the inverse problem boundary value problem for (1.1), (1.2) was studied for Weyl function, the uniqueness of solution of inverse problem for the spectral data formed by the eigenvalues and normalizing numbers was shown.

Here we deal with boundary value problem (1.1), (1.2). Let <^(x, A) and -0(x, A) be the solutions of boundary value problem (1.1), (1.2) satisfying the initial conditions

p(0,A) = W(0,A) = 0 (1.4)

and

^(n,A) = 0,^'(n,A) = 1. (1.5)

Denote

A(A) = W[p(x, A), ^(x, A)] = p(x, A)^'(x, A) - </(x, A)^(x, A). (1.6)

Function A(A) is called the characteristic function of problem (1.1), (1.2). Substituting x = 0 and x = n into (1.6), we get

A(A) = p(n,A) = ^'(0,A). (1.7)

Lemma 1. The eigenfunctions yi(x,Ai) and y2(x, A2) corresponding to different eigenvalues Ai = A2 are orthogonal.

Proof. Since yi(x, Ai) and y2(x, A2) are eigenfunctions of problem (1.1), (1.2), we get

—y"(x, Ai) + q(x)yi(x, Ai) = Aip(x)yi(x, Ai),

—A2) + q(x)yi(x A2) = ^p^lOx A2).

Multiplying these identities by y2(x,A2 ) and —yi(x,Ai), respectively, and summing up, we obtain

d dx

Integrating from 0 to n and using the condition (1.2), we have

— {< ^^ ^yi^ >} = (A2 — A2M^yi^

¡•n

22

(Ai — A2W p(x)yi(x,Ai)y2(x,A2)dx = 0. Jo

Since Ai = A2,

/ p(x)yi(x, Ai)y2(x, Ai)dx = 0. Jo

□

Corollary 1. The eigenvalues of boundary value problem (1.1),(1.2) are real.

Lemma 2. The zeros An of characteristic function coincide with the eigenvalues of boundary value problem (1.1), (1.2). The functions <^(x, An) and ^(x,An) are eigenfunctions, and there exists a sequence such that

^(x,An)= ^n^(x,An), ^n = 0. (1.8)

Proof. 1) Let Ao be zero of A(A). Then, because of (1.6), — (x, Ao) = /5o<^(x, Ao) and the function — (x, Ao) and <^(x, Ao) satisfy boundary condition (1.2). Thus, A0 is an eigenvalue and — (x,Ao), <^(x, Ao) are associated eigenfunctions.

2) Let Ao be an eigenvalue of problem (1.1), (1.2) and let yo(x) be an associated eigenfunction. Then, yo(x) satisfies boundary condition (1.2). Clearly, yo(x) = 0. Without loss of generality, we let yo(0) = 1. Then yo(0) = 0 and therefore, yo(x) = <^(x, A). Hence, by (1.7), Ao(A) = 0. We have proved that for each eigenvalue there exists only one eigenfunction. □

Definition 1. The normalizing number of boundary value problem (1.1), (1.2) is described

as

P7T

an := / p(x)^2(x, An)dx. o

Lemma 3. The eigenvalues of boundary value problem (1.1), (1.2) are simple and

A (A) = 2Anan^n. (1.9)

Proof. Since <^(x, An) and —(x, A) are the solutions of this problem, the identities

—<p"(x, An) + q(x)^(x, An) = Anp(x)^(x, An), ——"(x, A) + q(x)—(x, A) = A2p(x)—(x, A) hold true. Multiplying them by —(x, A ) and —<^(x, An), respectively, and summing up, we get d

— {< <p(x, An),-(x, A) >} = (An — A2)p(x)^(x, An)^(x, A). Integrating from 0 to n and using the condition (1.2), we obtain:

jf p(xV(^ An)-(^ A) = A(^n2)- ^(A) . Since —(x, An) = ^n^(x, An) as A ^ An, by Lemma 3, we obtain

A(An) = 2Anan^n

where = -0(0, An). Thus, it follows that A(An) = 0. □

2. On the Eigenvalues of Problem (1.1), (1.2) as q(x) = 0

We denote by <^o(x, A) the solution to the equation — y" = A2p(x)y satisfying condition (1.4). It reads as:

^o(x,A)=2 +ypp^yjcos A^+(x)+2 —^p(xy)cos A^-(x) (2.1)

where ^±(x) = ±x\Jp(x) + a(1 ^ \Jp(x)).

We see that if (An)2 are the eigenvalues of problem (1.1), (1.2) as q(x) = 0, then An can be found via the equation <^o(n, A) = 0, that is, by the equation

Ao(A) = ^(1 + 1)cos (n) + ^(1 — 1)cos A^-(n) = 0 2 a 2 a

a — 1

cos (n) +--cos A^-(n) = 0. (2.2)

a + 1

It follows from (2.2) that

where

sup |hn| < x.

un |

n

Lemma 4. The roots of function A0(A) are isolated, i.e.,

i=0 I An - Aufc I = y > 0.

n=0

Proof. We argue by the contradiction. Assume the contrary, then there are two sequences {A^}, {Ak,2} of zeros of function A0 (A) such that

Ak,i = A0fc,2, Ak,i ^ x, A0fc,2 ^ x, lim ^ - A^) = 0. By Lemma 1, functions (x, A° ^ and (x, AO 2) are orthogonal:

0 = P (x) Po (x A°,0 ^o (x A0J dx

0

= p (x) ^o (x A0,0 [Vo (x A0,^ - ^o (x A0,0] dx+

+ / P (x) Po (x A0,i) dx =

Jo

pn pa

=/0 + / P (x) ^o (x, A0,0 dx ^ Io + / P (x) ^ (x, AO, J dx

a

2 ^o ) , _ r , a sin(2aA0,i

=Io + cos2 (A0,ix dx = Io + - -

2 4Ao,i '

where

pn

I0 = P (x) ^o A0,l) [^o A0,2) - ^o A°,0] dx.

o

Let us show that

lim I0 = 0.

fc—>oo

Indeed, by (2.1) and the estimate

V0,ix - cos A0,2

cos A01x - cos AO 2x| ^ C | AO 1 - AO 2| (C > 0)

we conclude that

|po (x, AOo ,i) - ^ (x, AO,2) | ^ C | AO, 1 - AO,21 (C > 0).

Thus, lim (x,Ak , J — (x, A°k,2)) =0 is valid uniformly for x G [0,n]. Passing to the

\0

limit in the inequality 0 ^ Ik + f — sin4:^0Afc'l) as k ^ œ, we have 0 ^ f. We arrive at the

k, 1

contradiction. □

3. Asymptotic Formulas of Eigenfunctions and Eigenvalues

Using representation for solution e(x,A) to equation (1.1) satisfying the initial conditions e(0, A) = 1, e!(0, A) = iA, we obtain the following integral representation for solution <^(x, A):

r M+(x)

<^(x, A) = ^o(x,A)+ / A(x,t)cos Atdt Jo

where K(x,.) G L(—(x)) and A(x,t) = K(x,t) — K(x, —t). Kernel A(x,t) processes the following properties:

» A(^+M) = 4 Jo' ^ (1 + ^fe)

ii) A(n, + 0) - A(n, Mn) - 0) = 1 ft ^ (l -Lemma 5. As |À| ^ the asymptotic formulas

I „|ImA|Ju+(x) > , *

y (x, A) = y (x, A) + O ^ = O (e|ImA|^+(x))

|A|

(3.1)

y' (x, A) = y0 (x, A) + O (>m^+(x)) = O (|A| e|ImA|^+(x))

|ImA|(M+(n)-M+(x)) \ / IImA|(^+(n)-^+(x)) '

e1 v > ^ v>) \ e

^ (x, A) = ^o (x, A) + O ( -—2- = O

|A|2 ; ~ V IA |

|ImA|(M+(n)-U+(x))'

(3.2)

^ (x, A) = ^0 (x, A) + O I e-A- = O (e|ImA^+(n)--+(x)))

hold true uniformly with respect to x G [0,n].

Proof. The standard method of variations of an arbitrary constants leads us to the following integral equation for the solution ^ (x, À) :

r-x

<^(x,À) = (x, À) + / g(x,t; À)q(i)^(i, À)dt (3.3)

Jo

where

1 ( 1 1 \ sinÀQa+(x) - ^+(t)) +

g(x t; À) =_ l VPM + TW))-À-+

+ 1_(_____O sinÀQu-(x) - ^+(t))

+ nVP(xy vW À (.)

and <^0(x, À) is the solution of equation (1.1) as q(x) = 0 satisfying the conditions (1.4). Denote

a (À)= max (|p(x,À)| e-|ImA|^+ (n)) .

Since

|sinÀ^+ (x)| ^ e|ImA|^+(x), |cos À^+(x)| ^ e|ImA|^+(x)

and

|g (x,t; À)| ^ ^e|ImA|(^+(x)-^+(i)),

IAI

by (3.3) we get that for |A| ^ 1 and x G [0,n],

and therefore

y^y f'X

|y(x, A)| e-|ImA|^+(n) ^ Ci + IÄ2a (A) y |q(t)I dt,

C2

a (A) ^ Ci + ^a (A)

and |g (x,t; À)| ^ e|ImA|(M+(x)-M+(t)) we conclude that

For sufficiently large |A| it yields a (A) = O(1), i.e. <^(x, A) = O (elImA^+(x)). By this identity

^ (x, A) = ^o (x, A) + O

e|ImA|^+(x)

Differentiating (3.3), we calculate:

r-x

v' (x,A) = vO (x,A) + g(x,x; A)q(x)v (x,A)+ / gX(x,t; A)dt (3.5)

o

where by (3.4), g(x,x; A) = 0 and

(3.6)

gX(x,t;A) ^/p^x) + cos A(^+(x) — (t)) +

+ ypp^xy— ^p^y)cos A(^-(x) — ^+(t)).

Substituting the identity v(x, A) = O ^e|ImA|M+(x) j into the right hand side of (3.5), we arrive at (3.1). In the same way one can get (3.2). □

Theorem 1. Boundary value problem (1.1),(1.2) has a countable set of simple eigenvalues

{AnWi-

d k

An = An + Ao + ^, (An ^ 0) (3.7)

An n

where An are the zeros of the function

Ao(A) = 1 ( 1 + - ) cos A^+(n) + 1 (1 — - J cos A^-(n), (3.8)

2 y a ) 2 y a J

{An}2 are the eigenvalues of problem (1.1),(1.2) as q(x) = 0,

d =_h+ sin An^+(n) + h~ sin An^- (n)_ (3 9)

n 2(1 + a)p+(n)smAn^+ (n) + 2(1 — a)Mn)sm( . )

is a bounded sequence and kn G l2.

Proof. Since A (A) = A) is the characteristic function of boundary value problem (1.1), (1.2), we have

pir

A(A) = A0(A) + g(n,t; A)q(t)v(t, A)dt. (3.10)

Jo

By (3.1) we obtain

a /xn , + sinAu+(n) , sinAu-(n) A(A) = Ao(A) + h+-^^ + h--^^ + Ko(A), (3.11)

where

h± = ± ± ji q(t)dt + 1(a ± 1) J^ q(t)dt, (3.12)

and

1

Ko(A) =tV(1 + aW cos A(2^+(i) — ^+(n))q(i)di+ 4A Jo

1 /""

(1 — aW cos A(2ju+(t) — ^-(n))q(t)dt+ 4A Jo

+ 4A (1 + a) / cos A(2^+(t) — ^+(n))q(i)di+

1 / 1 \ rn / g|ImA|^+(n) '

+ 4A (,1 — a) J cos A(^+(n) + p-(t) — M+(t))q(t)dt + O I |A|2

(3.13)

a

We denote Gj = {A : |A — ^ 8}, where 8 is a sufficiently small positive number 8 < 2 (see lemma 5). Let us show that

|Ao(A)| ^ (je|ImA|^+(n), A G Gj, (j > 0. (3.14)

We have |cos A^+(n)| ^ G«se|ImA|^+(n), A G Gj. Then, using (3.8), we get (3.14). Further, by (3.11) we arrive at

|A(A)| ^ (je|ImA|M+(n), A G Gj, Cj > 0. (3.15)

On the other hand, by (3.11) we obtain

/e|ImA|M+(n)\

A(A) — Ao(A) = Of |A| j, |A| —to. (3.16)

Consider the contour rn = {A : |A| = |Ani + 2} (n =1, 2,...). Enlarging unboundedly contour rn, for sufficiently large n by (3.14) and (3.16) we have

|A(A) — Ao(A)| ^ |Ao(A)|, A G rn . (3.17)

Applying now Rouche's theorem, we see that the number of zeros of Ao(A) inside rn coincides with the number of zeros of A(A) = {A(A) — Ao(A)} + Ao(A). We apply the Rouche's theorem to the circle Yn(8) = {A : |A — An| ^ 8} and conclude that for sufficiently large n, there exist only one zero An of the function A(A) in Yn(8). Since 8 > 0 is arbitrary, then

An = An + £n, £n = o(1), n — to. (3.18)

Substituting (3.18) into (3.11) and taking into consideration the relations

0 1 1 1 1

Ao(A^) = -(1 + -) cos A») + -(1 - -) cos A>-(n) = 0, 2 a 2 a

we get

where

sin(n) ~ (n), cosen^+(n) ~ 1, n ^ to,

dn I T i kn in 1 c\\

= Â^TT" + Än^ dn + Än^ (3.19)

n ' n n ' n n ' n

h+ sin An^+(n) + h sin An^ (n)

n i(1 + a)ï+(n)sin Anï+(n) + i(1 - a)ï-(n) sin Anï-(n) kn = ko (An + £n) and

h+ï+(n) cos An^+(n) + h-ï-(n) cos An^-(n)

dn

2(1 + a)ï+(n)sin Anï+(n) + 1 (1 - a)ï-(n) sin An^-(n)'

Since A0 +£ = O(1), Ao£+n£ = o(n), n —^ to, we have that dn, dn are bounded, kn G /2 and (3.19) implies

£n = O(—), n — to. n

Using (3.19) once more, we can obtain more precisely that

d k

£n = T"^ + —, kn G /2, n — +to, (3.20)

n Aon n n 2

where kn = M+(n) kn + O(n), n — to. The prove is complete. □

4. Spectral Expansion Formula

Theorem 2. 1) The system of eigenfunftions |^(x, An)}ra^1 of boundary value problem (1.1), (1.2) is complete in L2p(0,n);

2) If f (x) is an absolutely continuous function on the segment [0,n], and f '(0) = f (n) = 0, then

oo

f (x) = 5^ «n^^ (4.1)

n=1

where

1 r

an = — f (t)v(t, A„)p(t)dt, (4.2)

®n J 0

and series (4.1) converges uniformly on [0,n];

3) For f G ¿2,p(0,n) series (4.1) converges in ¿2,p(0,n), moreover, the Parseval identity

oo

f»7T

|f (x)|2 p(x)dx = ^ an |a„|2 (4.3)

-70 n=i

holds.

Proof. Let -0(x, A) be a solution of equation (1.1) under the initial conditions (1.5). Denote

G(x,t; A) = —( A) A), x<" (4.4)

v ' ' ; A(A) \ v(x,A)^(i,A), t<x v 7

and let us consider the function

i'TT

Y(x, A) = / p(t)f (t)G(x,t; A)dt (4.5)

o

which is a solution to the boundary value problem

—Y''(x, A) + p(x)Y(x, A) = A2p(x)Y(x, A) — f (x)p(x), (4.6)

Y'(0, A) = 0, Y(n, A) = 0.

Using (1.9), we obtain

1 r

ResY(x, A) = —^^v(x,AnW p(t)f (t)v(t, An)dt. (4.7)

A—An 2anAn ,/0

Let f (x) G L2,p(0,n) be such that

i'TT

/ p(t)f (t)v(t, An)dt = 0 n = 1, 2, 3,....

o

Then, from (4.7), we have ResY(x, A) = 0. Hence, for fixed x G [0,n], function Y(x, A) is entire

A—An

with respect to A. On the other hand, substituting (3.1), (3.2) and (3.15) into (4.5), we see that that for a fixed 8 > 0 and a sufficiently large A* > 0 :

|Y(x, A)| ^ CA, A G , |A| > A*.

Using the maximum principle for module of analytic functions and Liouville theorem, we conclude that Y(x, A) = 0. This fact and (4.6) imply that f (x) = 0 a.e. on [0,n]. Thus, statement 1) of the theorem is proved.

Let f G AC[0,n]. We rewrite function Y(x, A) as

1 r ?x

Y(x, A) = - pA^I^(x,A)y (VM) + ?(%(*, A)) f (t)dt+

fTT

l.ll

+^(x,AW (-^"(t,A) + g(t)^(t,A)) f (t)dt .

Integrating by parts the term with the second-order derivatives and taking into consideration the conditions f'(0) = 0, f (n) = 0, we obtain

Y(x, A) = M — A2 (Zi(x, A) + Z2(x, A)), (4.8)

where

Zi(x,A) = AA) Here g(t) = f'(t),

Z (x, A) = 1

^(x,AW #(%'(t,A)dt + ^(x,AW g(i)^'(i,A)di

./o Jx

rx pn

#z,A) / p(t,A))f (t)q(t)dt + p(x,A) / ^(i,A))f (t)q(t)dt - p(x,A)f (n)

ox

A(A)

We consider the contour integral

1

1N (x) =- AY(x, A)dA,

2ni ,/r

^ 1 n

where rn = {A : |A| = |AN| + 2} is a counter-clockwise oriented contour. By means of the residue theorem we have

N N

In(x) = 2 V ResY(x, A) = V «n^(x, An) (4.9)

A—A

n—i n—i

where

1 r

an = — p(t)f (t)v(t, An)dt. «W 0

On the other hand, taking into consideration (4.8), we have

1 f 1

In(x) = f (x) — — — (Zi(x, A) + Z2(x,A))dA. (4.10)

rN A

Comparing (4.9) and (4.10), we obtain

N

( x) = a

f (x) = an^(x, An) + Cw(x),

n=i

where

11

Cn(x) = — -(Zi(x, A) + Z>(x, A))dA.

rN A

Therefore, in order to prove the item 2) of the theorem, it suffices to show that

lim max |£n(x)| = 0 (4.11)

From (3.1), (3.2) and (3.15) it follows that for fixed 8 > 0 and sufficiently large A* > 0

C2

max |Z2(x,A)| , A G , |A| ^ A*,C2 > 0. (4.12)

|A|

Let us show that

lim max |Zi(x,A)| = 0. (4.13)

o^n 1

A€Gä

x

7T

First we suppose that g(t) is absolutely continuous on [0,n]. In this case, integration by parts gives

Zi(x,À) = --r^ <mx,À) f ^(i,À)g'(i)di + p(x,À) f ^(i,À)g'(i)di

A(A)k ./0

Therefore, similarly to Z2(x, A), we have

max |Zi(x,A)| ^ ^, A G G«s, |A| ^ A*,C1 > 0.

O^x^n |A|

In the general case, we fix e > 0 and choose an absolutely continuous function g£(i) such that

i'TT

/ |ge(i) - g(t)| dt < e.

0

Then, using estimates (3.1), (3.2) and (3.15), one can find A** > 0 such that for A G G«s, |A| ^ A** the relation

Zi(x, A)

1

A(A)

+

A(A)

-(x,A) / ^'(i,A)(g(i) - ge(t))dt + ^(x,A)/ -'(t,A)(g(t) - ge(t))dt

-(x,A) / p(t,A)^(t)dt - ^(x,AW -M)^(t)dt

./0 Jx

+

yields

Therefore,

max |Zi(x, A)| ^ G Г |ge(t) - g(t)| dt + ^ < G + ^, A G Gô, |A| ^ A* 0 | A| | A|

lim max |Z1(x,A)| ^ Ge.

A€Gä

Since e is an arbitrary positive number, we arrive at identity (4.13). Relations (4.12), (4.13) immediately imply (4.11), and thus, statement 2) of the theorem is proved.

System of eigenfunction {<^(x, An)}n^1 is complete and orthogonal in L2,p(0,n). Therefore, it forms the orthogonal basis in L2,p(0,n) and Parseval identity (4.3) is valid. □

5. WEYL SOLUTION, WEYL FUNCTION

Let $(x,À) be the solution of equation (1.1) satisfying the conditions $'(0,À) = 1, $(n,À) = 0. Denote by C(x, À) the solution of equation (1.1) satisfying the initial conditions C(0, À) = 0, C'(0, À) = 1. Then, the solution ^(x, À) can be represented as

or

-(x, A) = -0(0, A)p(x, A) + A(A)C (x, A)

= G (x, A) + , Д , ^(x,A).

A(A)

Denote

It is clear that

M (A)

A(A)

-(0, A) A(A) .

(5.1)

(5.2)

(5.3)

(5.4)

$(x, A) = C(x, A) + M(A)p(x, A).

Functions $(x, A) and M(A) = $(0, A) are respectively called the Weyl solution and the Weyl function of boundary value problem (1.1), (1.2). The Weyl function is a meromorphic function

x

0

x

1

having simple poles at points An being the eigenvalues of boundary value problem (1.1), (1.2 Relations (5.2), (5.4) yield

A)

$(x, A)

A(A)

It can be shown that

< <^(x, A), $(x, A) >= 1.

5.5)

(5.6)

Theorem 3. If M(A) = M(A), then L = L; that is, the boundary value problem (1.1), (1.2) is uniquely determined by the Weyl function.

Proof. We introduce the matrix P(x, A) = [Pj(x, A)]. ?—i 2 by the formula

P (x, A)

<^(x,A) $(x,A) <^'(x,A) $'(x,A)

<^(x,A) $(x, A) p'(x,A) $'(x, A)

By (5.7) we have

(5.7)

or

<^(x, A) = Pii(x, A)^(x, A) + Pi2(x, A)<//(x, A), $(x, A) = Pii(x, A)S(x, A) + Pi2(x, A)$'(x, A),

Pii(x, A) = p(x, A)$'(x, A) - $(x, A)p'(x, A), Pi2(x, A) = -<p(x, A)$(x, A) + $(x, A) £(x, A).

(5.8)

(5.9)

Taking into consideration equations (5.5) and (5.9), we substitute (5.4) into (5.9) to obtain

p(x, A)(^'(x, A) - ^'(x, A)) - ^(x, A)(^ '(x, A) - ^(x, A)) i

Pii(x, A) = 1 + ^

P12(x, A) = ÄiÄ) A)^(X, A) - AMx A)

(5.10)

By (3.1), (3.2), (3.15) and equation (5.10) we get

(5.11)

lim max iPnfx, A) — 11 = lim max |Pi2(x,A)| = 0. Hence, if we take into consideration equations (5.4) and (5.9), we get

Pii(x, A) = v(x, A)C'(x, A) — C(x, A)p'(x, A) + (M(A) — M(A))v(x, A)p'(x, A) Pi2(x, A) = C(x, A)q(x, A) — C(x, A)v(x, A) + (M(A) — M(A))v(x, A)q(x, A).

Therefore if M(A) = M(A), then Pn(x,A) and Pi2(x,A) are entire functions for each fixed x. It can be easily seen from (5.11) that Pn(x,A) = 1 and Pi2(x, A) = 0. Substituting it into

(5.8), we get v(x,A) = q(x, A) and $(x,A) = $(x,A) for each x and A. Hence, we arrive at q(x) = q (x). □

Theorem 4. The expression

M (A)

2Aoao(Ao - A)

+ £

n=1

an (An - A2)

(5.12)

holds true.

1

1

Proof. Using (1.9), we calculate

where A(A) = dxk(A). Taking into account the last identities, in accordance with (5.3) we calculate:

ResM (A) = ^nl = = 1 . (5.13)

A=An A (A„) k(A«) 2Anan

Using (3.2), (3.15) and (5.3), we have

Thus, we get

|M(A)| ^ CA, A e G,.

lim |M(A)| = 0. (5.14)

Now, let us consider the contour integral

Jn(A) = i M(4d^, A e /ntrw, 2rn JrN y - A

where the contour rN = : = |AN| + 2} is passed counter-clockwise.

Owing to (5.14), we have lim JN(A) = 0. On the other hand, by the residue theorem, the

N ^^

identity A-n = — An and (5.13), we have

N 1

Jn(A) = M(A) + E 2A„a„(A„ — A)

n=-N

N

M (A)+2^a (l_A)+E

2Aoao(Ao — A) «=1 ««(An — A2) and as N ^ to, we arrive at (5.12). □

Theorem 5. If An = An, an = an for all n e Z, then L = L. That is, problem (1.1), (1.2) is uniquelly determined by its spectral data.

Proof. Since An = An, an = an for all n e Z, considering formula (5.12), we have M(A) = M(A). Using Theorem 9, we arrive at L = L. □

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Khanlar Residoglu Mamedov, Science and Letter Faculty, Mathematics Department, Mersin University, 333343, Mersin, Turkey E-mail: [email protected]

Done Karahan,

Science and Letters Faculty,

Mathematics Department,

Harran University,

Sanliurfa, Turkey

E-mail: [email protected]