Open Access

On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition

Boundary Value Problems20102010:171967

DOI: 10.1155/2010/171967

Received: 9 April 2010

Accepted: 22 May 2010

Published: 21 June 2010

Abstract

An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq1_HTML.gif with a linear spectral parameter in the boundary condition. The scattering data of the problem are defined and a new fundamental equation is derived, which is different from the classical Marchenko equation. With help of this fundamental equation, in terms of the scattering data, the potential is recovered uniquely.

1. Introduction

We consider inverse scattering problem for the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ1_HTML.gif
(1.1)
with the boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ2_HTML.gif
(1.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq2_HTML.gif is a spectral parameter, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq3_HTML.gif is a real-valued function satisfying the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ3_HTML.gif
(1.3)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq4_HTML.gif is a positive piecewise-constant function with a finite number of points of discontinuity, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq5_HTML.gif are real numbers, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq6_HTML.gif

The aim of the present paper is to investigate the direct and inverse scattering problem on the half-line https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq7_HTML.gif for the boundary value problem (1.1)–(1.3). In the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq8_HTML.gif , the inverse problem of scattering theory for (1.1) with boundary condition not containing spectral parameter was completely solved by Marchenko [1, 2], Levitan [3, 4], Aktosun [5], as well as Aktosun and Weder [6]. The discontinuous version was studied by Gasymov [7] and Darwish [8]. In these papers, solution of inverse scattering problem on the half-line https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq9_HTML.gif by using the transformation operator was reduced to solution of two inverse problems on the intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq11_HTML.gif . In the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq12_HTML.gif , the inverse scattering problem was solved by Guseĭnov and Pashaev [9] by using the new (nontriangular) representation of Jost solution of (1.1). It turns out that in this case the discontinuity of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq13_HTML.gif strongly influences the structure of representation of the Jost solution and the fundamental equation of the inverse problem. We note that similar cases do not arise for the system of Dirac equations with discontinuous coefficients in [10]. Uniqueness of the solution of the inverse problem and geophysical application of this problem for (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq14_HTML.gif were given by Tihonov [11] and Alimov [12]. Inverse problem for a wave equation with a piecewise-constant coefficient was solved by Lavrent'ev [13]. Direct problem of scattering theory for the boundary value problem (1.1)–(1.3) in the special case was studied in [14].

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq15_HTML.gif in (1.1) with the spectral parameter appearing in the boundary conditions, the inverse problem on the half-line was considered by Pocheykina-Fedotova [15] according to spectral function, by Yurko [1618] according to Weyl function, and according to scattering data in [19, 20]. This type of boundary condition arises from a varied assortment of physical problems and other applied problems such as the study of heat conduction by Cohen [21] and wave equation by Yurko [16, 17]. Spectral analysis of the problem on the half-line was studied by Fulton [22].

Also, physical application of the problem with the linear spectral parameter appearing in the boundary conditions on the finite interval was given by Fulton [23]. We recall that inverse spectral problems in finite interval for Sturm-Liouville operators with linear or nonlinear dependence on the spectral parameter in the boundary conditions were studied by Chernozhukova and Freiling [24], Chugunova [25], Rundell and Sacks [26], Guliyev [27], and other works cited therein.

This paper is organized as follows. In Section 2, the scattering data for the boundary value problem (1.1)–(1.3) are defined. In Section 3, the fundamental equation for the inverse problem is obtained and the continuity of the scattering function is showed. Finally, the uniqueness of solution of the inverse problem is given in Section 4.

For simplicity we assume that in (1.1) the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq16_HTML.gif has a discontinuity point:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq17_HTML.gif .

The function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ5_HTML.gif
(1.5)

is the Jost solution of (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq18_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq19_HTML.gif .

It is well known [9] that, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq20_HTML.gif from the closed upper half-plane, (1.1) has a unique Jost solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq21_HTML.gif which satisfies the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ6_HTML.gif
(1.6)
and it can be represented in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ7_HTML.gif
(1.7)
where the kernel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq22_HTML.gif satisfies the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ8_HTML.gif
(1.8)
and possesses the following properties:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ9_HTML.gif
(1.9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ10_HTML.gif
(1.10)
In addition, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq23_HTML.gif is differentiable, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq24_HTML.gif satisfies (a.e.) the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ11_HTML.gif
(1.11)
Denote that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ12_HTML.gif
(1.12)

According to Lemma 2.2 in Section 2, the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq25_HTML.gif has only a finite number of simple roots in the half-plane https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq26_HTML.gif ; all these roots lie in the imaginary axis. The behavior of this boundary value problem (1.1)–(1.3) is expressed as a self-adjoint eigenvalue problem.

We will call the function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ13_HTML.gif
(1.13)

the scattering function for the boundary value problem (1.1)–(1.3), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq27_HTML.gif denotes the complex conjugate of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq28_HTML.gif .

We denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq29_HTML.gif   the normalized numbers for the boundary problem (1.1)–(1.3):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ14_HTML.gif
(1.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq30_HTML.gif . It turns out that the potential https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq31_HTML.gif in the boundary value problem (1.1)–(1.3) is uniquely determined by specifying the set of values https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq32_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq33_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq34_HTML.gif The set of values is called the scattering data of the boundary value problem (1.1)–(1.3). The inverse scattering problem for boundary value problem (1.1)–(1.3) consists in recovering the coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq35_HTML.gif from the scattering data.

The potential https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq36_HTML.gif is constructed by slightly varying the method of Marchenko. Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ15_HTML.gif
(1.15)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ16_HTML.gif
(1.16)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq37_HTML.gif

We can write out the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ17_HTML.gif
(1.17)

for the unknown function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq38_HTML.gif . The integral equation is called the fundamental equation of the inverse problem of scattering theory for the boundary problem (1.1)–(1.3). The fundamental equation is different from the classic equation of Marchenko and we call the equation the modified Marchenko equation. The discontinuity of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq39_HTML.gif strongly influences the structure of the fundamental equation of the boundary problem (1.1)–(1.3). By Theorem 4.1 in Section 4, the integral equation has a unique solution for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq40_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq41_HTML.gif Solving this equation, we find the kernel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq42_HTML.gif of the special solution (1.7), and hence according to formula (1.10) it is constructed the potential https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq43_HTML.gif

We show that formula (1.7) is valid for (1.1). For this, let us give the algorithm of the proof in [9]. For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq44_HTML.gif let us consider the integral equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ18_HTML.gif
(1.18)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ19_HTML.gif
(1.19)

while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq46_HTML.gif are solutions of (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq47_HTML.gif , satisfying the initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq49_HTML.gif

It is not hard to show that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq50_HTML.gif satisfies the formula
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ20_HTML.gif
(1.20)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ21_HTML.gif
(1.21)
Substituting the expression (1.7) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq51_HTML.gif in the integral equation (1.18) and using formula (1.20) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq52_HTML.gif after elementary operations, the following integral equations for the kernel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq53_HTML.gif are obtained:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ22_HTML.gif
(1.22)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq55_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ23_HTML.gif
(1.23)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq56_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq57_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ24_HTML.gif
(1.24)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq58_HTML.gif .

The solvability of these integral equations is obtained through the method of successive approximations. By using integral equations (1.22)–(1.24) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq59_HTML.gif equalities (1.9), (1.10) are obtained. By substituting the expressions for the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq61_HTML.gif in (1.1), it can be shown that (1.11) holds.

2. The Scattering Data

For real https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq62_HTML.gif the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq64_HTML.gif form a fundamental system of solutions of (1.1) and their Wronskian is computed as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq65_HTML.gif . Here the Wronskian is defined as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq66_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq67_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq68_HTML.gif be the solution of (1.1) satisfying the initial condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ25_HTML.gif
(2.1)

The following assertion is valid.

Lemma 2.1.

The identity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ26_HTML.gif
(2.2)
holds for all real https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq69_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ27_HTML.gif
(2.3)
with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ28_HTML.gif
(2.4)

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq70_HTML.gif is called the scattering function of the boundary value problem (1.1)–(1.3).

Lemma 2.2.

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq71_HTML.gif may have only a finite number of zeros in the half-plane https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq72_HTML.gif . Moreover, all these zeros are simple and lie in the imaginary axis.

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq73_HTML.gif for all real https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq74_HTML.gif , the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq75_HTML.gif is the possible real zero of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq76_HTML.gif . Using the analyticity of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq77_HTML.gif in upper half-plane and the properties of solution (1.7) are obtained that zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq78_HTML.gif form at most countable and bounded set having https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq79_HTML.gif as the only possible limit point.

Now let us show that all zeros of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq80_HTML.gif lie on the imaginary axis. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq82_HTML.gif are arbitrary zeros of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq83_HTML.gif . We consider the following relations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ29_HTML.gif
(2.5)
Multiplying the first of these relations by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq84_HTML.gif and the second by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq85_HTML.gif , subtracting the second resulting relation from the first, and integrating the resulting difference from zero to infinity, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ30_HTML.gif
(2.6)
On the other hand, according to the definition of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq86_HTML.gif , the following relation holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ31_HTML.gif
(2.7)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ32_HTML.gif
(2.8)
This formula yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ33_HTML.gif
(2.9)
Thus, using (2.6) and (2.9) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ34_HTML.gif
(2.10)

Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq87_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq88_HTML.gif In particular, the choice https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq89_HTML.gif at (2.10) implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq90_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq91_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq92_HTML.gif . Therefore, zeros of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq93_HTML.gif can lie only on the imaginary axis. Now, let us now prove that function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq94_HTML.gif has zeros in finite numbers. This is obvious if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq95_HTML.gif , because, under this assumption, the set of zeros cannot have limit points. In the general case, since we can give an estimate for the distance between the neighboring zeros of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq96_HTML.gif it follows that the number of zeros is finite (see [2, page 186]).

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ35_HTML.gif
(2.11)

These numbers are called the normalized numbers for the boundary problem (1.1)–(1.3).

The collections https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq97_HTML.gif are called the scattering data of the boundary value problem (1.1)–(1.3). The inverse scattering problem consists in recovering the coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq98_HTML.gif from the scattering data.

3. Fundamental Equation or Modified Marchenko Equation

From (1.9), (1.10), it is clear that in order to determine https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq99_HTML.gif it is sufficient to know https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq100_HTML.gif . To derive the fundamental equation for the kernel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq101_HTML.gif of the solution (1.7), we use equality (2.2), which was obtained in Lemma 2.1. Substituting expression (1.7) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq102_HTML.gif into this equality, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ36_HTML.gif
(3.1)
Multiplying both sides of relation (3.1) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq103_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq104_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq105_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq106_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq107_HTML.gif at the right-hand side we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ37_HTML.gif
(3.2)
Now we will compute the integral https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq108_HTML.gif . By elementary transforms we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ38_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq109_HTML.gif . Thus we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ39_HTML.gif
(3.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq110_HTML.gif is the Dirac delta function.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq111_HTML.gif , similarly we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ40_HTML.gif
(3.5)
Consequently, (3.2) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ41_HTML.gif
(3.6)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ42_HTML.gif
(3.7)
Let us show that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq112_HTML.gif the last expression in the sum equals zero. We note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq114_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq115_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ43_HTML.gif
(3.8)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq116_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq117_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq118_HTML.gif and hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ44_HTML.gif
(3.9)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq119_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq120_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq121_HTML.gif and hence, for this case, the inequality holds.

Therefore, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq122_HTML.gif (3.2) takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ45_HTML.gif
(3.10)
On the left-hand side of (3.1) with help of Jordan's lemma and the residue theorem and by taking Lemma 2.2 into account for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq123_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ46_HTML.gif
(3.11)
From the definition of normalized numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq124_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq125_HTML.gif in (2.11) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ47_HTML.gif
(3.12)
Thus, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq126_HTML.gif by taking (3.10) and (3.12) into account, from (3.2) we derive the relation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ48_HTML.gif
(3.13)
Consequently, we obtain for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq127_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ49_HTML.gif
(3.14)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ50_HTML.gif
(3.15)

Equation (3.14) is called the fundamental equation of the inverse problem of the scattering theory for the boundary problem (1.1)–(1.3). The fundamental equation is different from the classic equation of Marchenko and we call equation (3.14) the modified Marchenko equation. The discontinuity of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq128_HTML.gif strongly influences the structure of the fundamental equation of the boundary problem (1.1)–(1.3).

Thus, we have proved the following theorem.

Theorem 3.1.

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq129_HTML.gif , the kernel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq130_HTML.gif of the special solution (1.7) satisfies the fundamental equation (3.14).

By using the fundamental equation it is shown that the scattering function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq131_HTML.gif is continuous at all real points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq132_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ51_HTML.gif
(3.16)

It can be shown that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq133_HTML.gif tends to zero as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq134_HTML.gif and is the Fourier transform of some function in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq135_HTML.gif .

4. Solvability of the Fundamental Equation

Substituting scattering data into (3.15), we construct https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq137_HTML.gif . The fundamental equation (3.14) can be written in the more convenient form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ52_HTML.gif
(4.1)

We will seek the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq138_HTML.gif of (4.1) for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq139_HTML.gif in the same space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq140_HTML.gif .

We consider the operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq141_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq142_HTML.gif acting in the spaces https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq143_HTML.gif , respectively, by the rules
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ53_HTML.gif
(4.2)

which appear in the fundamental equation.

The operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq144_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq145_HTML.gif are compact in each space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq146_HTML.gif for every choice of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq147_HTML.gif . The proof of this fact completely repeats the proof of Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq148_HTML.gif which can be found in [2].

Substituting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq149_HTML.gif into (4.1), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ54_HTML.gif
(4.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ55_HTML.gif
(4.4)
In order to prove the solvability of the given fundamental equation, it suffices to verify that the homogenous equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ56_HTML.gif
(4.5)

has no nontrivial solutions in the corresponding space.

From the homogenous equation (4.5) we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ57_HTML.gif
(4.6)
and, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq150_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ58_HTML.gif
(4.7)
Using this equality in (4.5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ59_HTML.gif
(4.8)
or taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq151_HTML.gif we obtain the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ60_HTML.gif
(4.9)

from which (4.5) is obtained.

Theorem 4.1.

Equation (4.5) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq152_HTML.gif for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq153_HTML.gif .

To prove this theorem we need some of auxiliary lemmas.

Lemma 4.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq154_HTML.gif is a solution of the homogenous equation (4.5), then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq155_HTML.gif .

Proof.

In fact, the kernel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq156_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq157_HTML.gif can be approximated by a bounded function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq158_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq159_HTML.gif . By rewriting (4.5) in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ61_HTML.gif
(4.10)
we obtain an equation with a bounded function on the right-hand side, where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ62_HTML.gif
(4.11)
In the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq160_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ63_HTML.gif
(4.12)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ64_HTML.gif
(4.13)
Thus, the function on the right-hand side of (4.10) is bounded. Consequently, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq161_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ65_HTML.gif
(4.14)

and the series converges in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq162_HTML.gif as well as in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq163_HTML.gif ; that is, the solution of the homogenous equation (4.5) is bounded.

Corollary 4.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq164_HTML.gif is a solution of the homogenous equation (4.5), then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq165_HTML.gif .

Proof.

In fact, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq166_HTML.gif .

Thus, it suffices to investigate (4.5) in the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq167_HTML.gif .

Lemma 4.4.

The operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq168_HTML.gif acting in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq169_HTML.gif are nonnegative for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq170_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ66_HTML.gif
(4.15)
and equality is attained if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ67_HTML.gif
(4.16)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq171_HTML.gif is Fourier transform of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq172_HTML.gif .

Proof.

According to definitions of the operators https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq174_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ68_HTML.gif
(4.17)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq175_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ69_HTML.gif
(4.18)
by the Cauchy-Bunyakovskii inequality, or, equivalently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ70_HTML.gif
(4.19)
Therefore, the first term on the right-hand side of formula (4.17) is nonnegative. Since the second term is obviously nonnegative. Inequality (4.16) holds, with equality, if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ71_HTML.gif
(4.20)
This shows that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq176_HTML.gif is orthogonal to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq177_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq178_HTML.gif But then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ72_HTML.gif
(4.21)

which is possible if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq179_HTML.gif . Thus, inequality (4.15) holds, with equality for those functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq180_HTML.gif whose Fourier transform https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq181_HTML.gif satisfies conditions (4.16). The lemma is proved.

With the help of Lemmas 4.2 and 4.4, we obtain the proof of Theorem 4.1. It remains to show that the homogenous equation (4.5) has only the null solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq182_HTML.gif But, by Lemma 4.4 the Fourier transform https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq183_HTML.gif of any solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq184_HTML.gif of (4.5) satisfies the identity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq185_HTML.gif Hence, upon setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq186_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq187_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ73_HTML.gif
(4.22)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq188_HTML.gif is the Fourier transform of the function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ74_HTML.gif
(4.23)
which vanishes for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq189_HTML.gif identity (4.22) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_Equ75_HTML.gif
(4.24)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq190_HTML.gif Therefore, if (4.5) has nonzero solution, (4.24) has infinitely many linear independent solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq191_HTML.gif which in turn contradicts the compactness of the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq192_HTML.gif Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq193_HTML.gif

According to Theorems 3.1 and 4.1 the following result holds.

Theorem 4.5.

The scattering data uniquely determine the boundary value problem (1.1)–(1.3).

Proof.

To form the fundamental equation (3.14), it suffices to know the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq195_HTML.gif In turn, to find the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq196_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq197_HTML.gif it suffices to know only the scattering data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq198_HTML.gif . Given the scattering data, we can use formulas (3.15) to construct the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq199_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq200_HTML.gif and write out the fundamental equation (3.14) for the unknown function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq201_HTML.gif . According to Theorem 4.1, the fundamental equation has a unique solution. Solving this equation, we find the kernel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq202_HTML.gif of the special solution (1.7), and hence, according to formulas (1.9)-(1.10), it is constructed the potential https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq203_HTML.gif .

Remark 4.6.

In the case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F171967/MediaObjects/13661_2010_Article_899_IEq204_HTML.gif is a positive piecewise-constant with a finite number of points of discontinuity, similar results can be obtained.

Declarations

Acknowledgment

This research is supported by the Scientific and Technical Research Council of Turkey.

Authors’ Affiliations

(1)
Mathematics Department, Science and Letters Faculty, Mersin University

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Copyright

© Khanlar R. Mamedov. 2010

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