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

- KhanlarR Mamedov
^{1}Email author

**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 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

is a positive piecewise-constant function with a finite number of points of discontinuity, are real numbers, and

The aim of the present paper is to investigate the direct and inverse scattering problem on the half-line for the boundary value problem (1.1)–(1.3). In the case , 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 by using the transformation operator was reduced to solution of two inverse problems on the intervals and . In the case , 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 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 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 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 [16–18] 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.

is the Jost solution of (1.1) when where .

According to Lemma 2.2 in Section 2, the equation has only a finite number of simple roots in the half-plane ; 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.

*the scattering function* for the boundary value problem (1.1)–(1.3), where
denotes the complex conjugate of
.

where
. It turns out that the potential
in the boundary value problem (1.1)–(1.3) is uniquely determined by specifying the set of values
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
from the scattering data.

for the unknown function
. 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
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
.
Solving this equation, we find the kernel
of the special solution (1.7), and hence according to formula (1.10) it is constructed the potential

while and are solutions of (1.1) when , satisfying the initial conditions and

The solvability of these integral equations is obtained through the method of successive approximations. By using integral equations (1.22)–(1.24) for equalities (1.9), (1.10) are obtained. By substituting the expressions for the functions and in (1.1), it can be shown that (1.11) holds.

## 2. The Scattering Data

For real the functions and form a fundamental system of solutions of (1.1) and their Wronskian is computed as . Here the Wronskian is defined as

The following assertion is valid.

Lemma 2.1.

The function
is called *the scattering function* of the boundary value problem (1.1)–(1.3).

Lemma 2.2.

The function may have only a finite number of zeros in the half-plane . Moreover, all these zeros are simple and lie in the imaginary axis.

Proof.

Since for all real , the point is the possible real zero of the function . Using the analyticity of the function in upper half-plane and the properties of solution (1.7) are obtained that zeros of form at most countable and bounded set having as the only possible limit point.

Here , In particular, the choice at (2.10) implies that , or , where . Therefore, zeros of the function can lie only on the imaginary axis. Now, let us now prove that function has zeros in finite numbers. This is obvious if , 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 it follows that the number of zeros is finite (see [2, page 186]).

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

The collections
are called *the scattering data* of the boundary value problem (1.1)–(1.3). The inverse scattering problem consists in recovering the coefficient
from the scattering data.

## 3. Fundamental Equation or Modified Marchenko Equation

where is the Dirac delta function.

If , then and hence, for this case, the inequality holds.

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
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 , the kernel of the special solution (1.7) satisfies the fundamental equation (3.14).

It can be shown that tends to zero as and is the Fourier transform of some function in .

## 4. Solvability of the Fundamental Equation

We will seek the solution of (4.1) for every in the same space .

which appear in the fundamental equation.

The operators are compact in each space for every choice of . The proof of this fact completely repeats the proof of Lemma which can be found in [2].

has no nontrivial solutions in the corresponding space.

from which (4.5) is obtained.

Theorem 4.1.

Equation (4.5) has a unique solution for each fixed .

To prove this theorem we need some of auxiliary lemmas.

Lemma 4.2.

If is a solution of the homogenous equation (4.5), then .

Proof.

and the series converges in as well as in ; that is, the solution of the homogenous equation (4.5) is bounded.

Corollary 4.3.

If is a solution of the homogenous equation (4.5), then .

Proof.

Thus, it suffices to investigate (4.5) in the space .

Lemma 4.4.

where is Fourier transform of the function .

Proof.

which is possible if and only if . Thus, inequality (4.15) holds, with equality for those functions whose Fourier transform satisfies conditions (4.16). The lemma is proved.

for all Therefore, if (4.5) has nonzero solution, (4.24) has infinitely many linear independent solutions which in turn contradicts the compactness of the operator Hence,

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 and In turn, to find the functions it suffices to know only the scattering data . Given the scattering data, we can use formulas (3.15) to construct the functions and write out the fundamental equation (3.14) for the unknown function . According to Theorem 4.1, the fundamental equation has a unique solution. Solving this equation, we find the kernel of the special solution (1.7), and hence, according to formulas (1.9)-(1.10), it is constructed the potential .

Remark 4.6.

In the case when 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

## References

- Marchenko VA:
**On reconstruction of the potential energy from phases of the scattered waves.***Doklady Akademii Nauk SSSR*1955,**104:**695-698.MATHMathSciNetGoogle Scholar - Marchenko VA:
*Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications*.*Volume 22*. Birkhäuser, Basel, Switzerland; 1986:xii+367.View ArticleGoogle Scholar - Levitan BM:
**On the solution of the inverse problem of quantum scattering theory.***Mathematical Notes*1975,**17**(4):611-624.MathSciNetView ArticleGoogle Scholar - Levitan BM:
*Inverse Sturm-Liouville problems*. VSP, Zeist, The Netherlands; 1987:x+240.MATHGoogle Scholar - Aktosun T:
**Construction of the half-line potential from the Jost function.***Inverse Problems*2004,**20**(3):859-876. 10.1088/0266-5611/20/3/013MATHMathSciNetView ArticleGoogle Scholar - Aktosun T, Weder R:
**Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation.***Inverse Problems*2006,**22**(1):89-114. 10.1088/0266-5611/22/1/006MATHMathSciNetView ArticleGoogle Scholar - Gasymov MG:
**The direct and inverse problem of spectral analysis for a class of equations with a discontinuous coefficient.**In*Non-Classical Methods in Geophysics*. Edited by: Lavrent'ev MM. Nauka, Novosibirsk, Russia; 1977:37-44.Google Scholar - Darwish AA:
**The inverse problem for a singular boundary value problem.***New Zeland Journal of Mathematics*1993,**22:**37-66.Google Scholar - Guseĭnov IM, Pashaev RT:
**On an inverse problem for a second-order differential equation.***Uspekhi Matematicheskikh Nauk*2002,**57**(3(345)):147-148.View ArticleGoogle Scholar - Mamedov KhR, Çöl A:
**On the inverse problem of the scattering theory for a class of systems of Dirac equations with discontinuous coefficient.***European Journal of Pure and Applied Mathematics*2008,**1**(3):21-32.MATHMathSciNetGoogle Scholar - Tihonov AN:
**On the uniqueness of the solution of the problem of electric prospecting.***Doklady Akademii Nauk SSSR*1949,**69:**797-800.MathSciNetGoogle Scholar - Alimov ŠA:
**A. N. Tihonov's works on inverse problems for the Sturm-Liouville equation.***Uspekhi Matematicheskikh Nauk*1976,**31**(6):84-88. English translation in*Russian Mathematical Surveys*, vol. 31, pp. 87–92, 1976MATHMathSciNetGoogle Scholar - Lavrent'ev MM Jr.:
**An inverse problem for the wave equation with a piecewise-constant coefficient.***Sibirskiĭ Matematicheskiĭ Zhurnal*1992,**33**(3):101-111, 219. translation in*Siberian Mathematical Journal*, vol. 33, no. 3, pp. 452–461, 1992MATHMathSciNetGoogle Scholar - Mamedov KhR, Palamut N:
**On a direct problem of scattering theory for a class of Sturm-Liouville operator with discontinuous coefficient.***Proceedings of the Jangjeon Mathematical Society*2009,**12**(2):243-251.MATHMathSciNetGoogle Scholar - Pocheykina-Fedotova EA:
**On the inverse problem of boundary problem for second order differential equation on the half line.***Izvestiya Vuzov*1972,**17:**75-84.Google Scholar - Yurko VA:
**On the reconstruction of the pencils of differential operators on the half-line.***Mathematical Notes*2000,**67**(2):261-265. 10.1007/BF02686257MATHMathSciNetView ArticleGoogle Scholar - Yurko VA:
**An inverse problem for pencils of differential operators.***Sbornik: Mathematics*2000,**191**(10):1561-1586. 10.1070/SM2000v191n10ABEH000520MATHMathSciNetView ArticleGoogle Scholar - Yurko VA:
*Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-Posed Problems Series*. VSP, Utrecht, The Netherlands; 2002:vi+303.View ArticleGoogle Scholar - Mamedov KhR:
**Uniqueness of the solution of the inverse problem of scattering theory for the Sturm-Liouville operator with a spectral parameter in the boundary condition.***Mathematical Notes*2003,**74**(1-2):136-140.MATHMathSciNetView ArticleGoogle Scholar - Mamedov KhR:
**On the inverse problem for Sturm-Liouville operator with a nonlinear spectral parameter in the boundary condition.***Journal of the Korean Mathematical Society*2009,**46**(6):1243-1254. 10.4134/JKMS.2009.46.6.1243MATHMathSciNetView ArticleGoogle Scholar - Cohen DS:
**An integral transform associated with boundary conditions containing an eigenvalue parameter.***SIAM Journal on Applied Mathematics*1966,**14:**1164-1175.MATHMathSciNetView ArticleGoogle Scholar - Fulton CT:
**Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions.***Proceedings of the Royal Society of Edinburgh. Section A*1980/81,**87**(1-2):1-34. 10.1017/S0308210500012312MathSciNetView ArticleGoogle Scholar - Fulton CT:
**Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions.***Proceedings of the Royal Society of Edinburgh. Section A*1977,**77**(3-4):293-308.MATHMathSciNetView ArticleGoogle Scholar - Chernozhukova A, Freiling G:
**A uniqueness theorem for the boundary value problems with non-linear dependence on the spectral parameter in the boundary conditions.***Inverse Problems in Science and Engineering*2009,**17**(6):777-785. 10.1080/17415970802538550MATHMathSciNetView ArticleGoogle Scholar - Chugunova MV:
**Inverse spectral problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions.**In*Operator Theory, System Theory and Related Topics (Beer-Sheva/Rehovot, 1997), Operator Theory Advances and Applications*.*Volume 123*. Birkhäuser, Basel, Switzerland; 2001:187-194.View ArticleGoogle Scholar - Rundell W, Sacks P:
**Numerical technique for the inverse resonance problem.***Journal of Computational and Applied Mathematics*2004,**170**(2):337-347. 10.1016/j.cam.2004.01.035MATHMathSciNetView ArticleGoogle Scholar - Guliyev NJ:
**Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions.***Inverse Problems*2005,**21**(4):1315-1330. 10.1088/0266-5611/21/4/008MATHMathSciNetView ArticleGoogle Scholar

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