The inverse scattering problem of some Schrödinger type equation with turning point
- Zaki FA El-Raheem^{1}Email author and
- Farouk A Salama^{1}
https://doi.org/10.1186/s13661-015-0316-6
© El-Raheem and Salama; licensee Springer. 2015
Received: 3 December 2014
Accepted: 12 March 2015
Published: 3 April 2015
Abstract
In this paper the inverse scattering problem is considered for a version of the one-dimensional Schrödinger equation with turning point on the half-line \((0, \infty)\). The scattering data of the problem is defined and the fundamental equation is derived. With the help of the derived fundamental equation, in terms of the scattering data, the potential is recovered uniquely.
Keywords
Schrödinger equation scattering function asymptotic formula contour integration main integral equation uniqueness theorem inverse scattering problemMSC
58C40 34L25 34B05 47A401 Introduction and preliminaries
1.1 Introduction
Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology, and other branches of the natural sciences. Inverse problems also play an important role in solving nonlinear evolution equations in mathematical physics. Interest in this subject has been increasing permanently because of the appearance of new important applications, and nowadays the inverse problem theory is developed intensively all over the world. The greatest success in spectral theory in general, and in particular in inverse spectral problems, has been achieved for the Sturm-Liouville operator \(y:=-y '' +q(x)y\), which also is called the one-dimensional Schrödinger operator.
The main results on inverse spectral problems appeared in the second half of the 20th century. We mention here the works by R Beals, G Borg, LD Faddeev, MG Gasymov, IM Gelfand, BM Levitan, N Levinson, VA Marchenko, and others (see [1] for details). An important role in the inverse spectral theory for the Sturm-Liouville operator was played by the transformation operator method (see [2, 3] and the references therein).
At present, other effective methods for solving inverse spectral problems have been created; among them we point out the method of spectral mappings connected with ideas of the contour integration method. This method seems to offer perspective for inverse spectral problems. The created methods allowed one to solve a number of important problems in various branches of the natural sciences.
The inverse scattering theory on the half-line and on the line was studied in [4–6], and others. In [7–11] KR Mamedov studied the inverse scattering theory on the half-line with spectral parameter contained in the boundary condition. Lately, there grew interest in the investigation of the boundary value problem by numerical methods; e.g. [12] presented an approximate construction of the Jost function for some Sturm-Liouville boundary value problem in the case \(\rho(x) = 1 \) by means of the collocation method; in addition, [13] is an application of spectral analysis of one-dimensional Schrödinger operators in a magnetic field. Also [14, 15] are applications of the discontinuous wave speed problem in a nonhomogeneous medium as in our case.
In the last 30 years there appeared many new areas for applications of inverse Sturm-Liouville problems, among them boundary value problems with discontinuity conditions inside the interval are connected with discontinuous material properties.
Many further applications were connected with the differential equation of the form \(y'' +q(x)y=r(x)y \) with turning points when the function \(r(x)\) has zeros and/or changes sign. For example, we have turning points connected with physical situations in which zeros correspond to the limit of motion of a wave mechanical particle bound by a potential field. Turning points appear also in elasticity, optics, geophysics, and other branches of natural sciences. Moreover, a wide class of differential equations with Bessel-type singularities and their perturbations can be reduced to differential equations having turning points; further inverse problems for equations with turning points and singularities help one to study blow-up solutions for some nonlinear integrable evolution equations of mathematical physics (see [16]).
Inverse problems of the Sturm-Liouville equation with turning points and singularities have been studied in [17–20], and other works.
The aim of the present paper is to investigate the inverse scattering problem on the half-line \([0, \infty)\) for some version of the one-dimensional Schrödinger equation with turning. In the case of \(\rho(x)= 1\), the inverse problem of scattering theory for (1.1) with boundary condition not containing a spectral parameter was completely solved by Marchenko [2, 21], Levitan [3, 22], Aktosun [23], and Aktosun and Weder [24]. The discontinuous version of ρ was studied by Gasymov and Levitan [25, 26], Darwish [27], and Gasymov and the author [19, 28]. In these papers, the solution of the inverse scattering problem on the half-line \([0, \infty)\) by using the transformation operator was reduced to the solution of two inverse problems on the intervals \([0,a ]\) and \([a, \infty)\). In the case \(\rho\neq1\) the inverse scattering problem was solved by the author, Guseinov, and Pashaev [19, 29] by using the new non-triangular representation of the Jost solution of (1.1). It turns out that in this case the discontinuity of the function \(\rho(x)\) strongly influences the structure of the representation of the Jost solution and the fundamental equation of the inverse problem.
The direct and inverse problem of (1.1) with \(y(0)-h y(0) = 0\) (see [21, 30]) has been solved earlier by the so-called spectral distribution function, while the problem (1.1) with \(y(0) = 0\) has been studied in the works [30, 31] by the inverse scattering method. Furthermore, the inverse scattering problem of the one-dimensional Schröodinger’s eigenvalue problem with a discontinuous coefficient was studied when \(y(0) = 0\) and \(y' (0)=0\) [27, 32] and [25]. It should be mentioned that the spectrum of the boundary value problem (1.1)-(1.2) has been previously investigated in [33] when \(\rho(x)>0\) and the boundary condition \(y(0)=0\) holds.
The present paper is organized as follows. Section 1 is an introduction and preliminaries in which we demonstrate some historical and scientific survey to inverse scattering problem. We introduce, from [34], the basic definitions and results that are needed in the subsequent investigation. In addition, the scattering data for the boundary value problem (1.1)-(1.2) are defined and some of its spectral properties are proved. In Section 2, the main integral equation of the inverse scattering problem is derived, by its scattering data. Finally, Section 3 is devoted to a proof of the uniqueness of both the main integral equation and the solution of the inverse scattering problem.
1.2 Preliminaries
Lemma 1.1
Proof
2 Formulation of the inverse scattering problem
2.1 Derivation of the main integral equation
Lemma 2.1
Proof
Lemma 2.2
Proof
Theorem 2.1
Proof
3 The uniqueness theorems
In this section we prove two theorems, first, the uniqueness theorem of the solution of the main integral equation (2.51) on the interval \((1,\infty)\), the second is the uniqueness of the inverse scattering problem of (1.1)-(1.2) by its scattering data (1.5).
3.1 The uniqueness theorem of the main integral equation
We prove the uniqueness of the solution of the integral equation (2.51) with respect to \(K(x,t)\).
Theorem 3.1
Proof
3.2 The uniqueness theorem of inverse scattering problem
Theorem 3.2
Proof
In Theorem 2.1, we have constructed the function \(H(x)\), \(x> 2 \), and used it to derive the main integral equation by the scattering data (3.13). From (2.22) we have \(q(x) = -2\frac {d}{dx}K(x,t)\), also from Theorem 3.1, the function \(K(x,t)\) is unique for \(1 < x <\infty\) and consequently the potential \(q(x)\), \(1 < x <\infty\), is unique. As a consequence of the uniqueness of \(q(x) \) the functions \(f(1,\eta)\) and \(f'(1,\eta)\) are uniquely defined.
Declarations
Acknowledgements
We are indebted to an anonymous referee for a detailed reading of the manuscript and useful comments and suggestions, which helped us improve this work. This work is supported by Office of Scientific Publishing Excellence of Alexandria University.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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