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This article is part of the series Proceedings of International Conference on Applied Analysis and Mathematical Modeling 2013.

Open Access Research

Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient

Khanlar R Mamedov and Ozge Akcay*

Author Affiliations

Mathematics Department, Science and Letters Faculty, Mersin University, Mersin, 33343, Turkey

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Boundary Value Problems 2014, 2014:110  doi:10.1186/1687-2770-2014-110


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/110


Received:30 November 2013
Accepted:25 April 2014
Published:13 May 2014

© 2014 Mamedov and Akcay; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In this paper, the inverse problem of recovering the coefficient of a Dirac operator is studied from the sequences of eigenvalues and normalizing numbers. The theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given.

MSC: 34A55, 34L40.

Keywords:
Dirac operator; inverse problem; necessary and sufficient condition

1 Introduction

In this paper, we consider the boundary value problem generated by the system of Dirac equations on the finite interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M1">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M2">View MathML</a>

(1)

with boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M3">View MathML</a>

(2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M4">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M6">View MathML</a> are real valued functions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M8">View MathML</a>, λ is a spectral parameter,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M9">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M10">View MathML</a>.

The inverse problem for the Dirac operator with separable boundary conditions was completely solved by two spectra in [1,2]. The reconstruction of the potential from one spectrum and norming constants was investigated in [3]. For the Dirac operator, the inverse periodic and antiperiodic boundary value problems were given in [4-6]. Using the Weyl-Titschmarsh function, the direct and inverse problems for a Dirac type-system were developed in [7,8]. Uniqueness of the inverse problem for the Dirac operator with a discontinuous coefficient by the Weyl function was studied in [9] and discontinuity conditions inside an interval were worked out in [10,11]. The inverse problem for weighted Dirac equations was obtained in [12]. The reconstruction of the potential by the spectral function was given in [13]. For the Dirac operator with peculiarity, the inverse problem was found in [14]. Inverse nodal problems for the Dirac operator were examined in [15,16]. In the case of potentials that belong entrywise to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M11">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M12">View MathML</a>, the inverse spectral problem for the Dirac operator was studied in [17], and in this work, not only the Gelfand-Levitan-Marchenko method but also the Krein method [18] was used. In the positive half line, the inverse scattering problem for the Dirac operator with discontinuous coefficient was analyzed in [19]. Besides, in a finite interval, for Sturm-Liouville operator inverse problem has widely been developed (see [20-22]). The inverse problem of the Sturm-Liouville operator with discontinuous coefficient was worked out in [23,24] and discontinuous conditions inside an interval were obtained in [25]. In the mathematical and physical literature, the direct and inverse problems for the Dirac operator are widespread, so there are numerous investigations as regards the Dirac operator. Therefore, we can mention the studies concerned with a discontinuity, which is close to our topic, in the references list.

In this paper, our aim is to solve the inverse problem for the Dirac operator with a piecewise continuous coefficient on a finite interval. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M14">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) be, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2). The quantities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) are called spectral data. We can state the inverse problem for a system of Dirac equations in the following way: knowing the spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) to indicate a method of determining the potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20">View MathML</a> and to find necessary and sufficient conditions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) to be the spectral data of a problem (1), (2). In this paper, this problem is completely solved.

We give a brief account of the contents of this paper in the following section.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23">View MathML</a> be solution of the system (1) satisfying the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M24">View MathML</a>

The solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23">View MathML</a> has an integral representation [26] as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M26">View MathML</a>

(3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M27">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M28">View MathML</a> is a quadratic matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30">View MathML</a> is the solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M31">View MathML</a>

(4)

Equation (4) gives the relation between the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30">View MathML</a> and the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20">View MathML</a> of (1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34">View MathML</a> be solutions of the system (1) satisfying the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M35">View MathML</a>

The characteristic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M36">View MathML</a> of the problem (1), (2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M37">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M38">View MathML</a> is the Wronskian of the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34">View MathML</a> and independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M41">View MathML</a>. The zeros <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M42">View MathML</a> of the characteristic function coincide with the eigenvalues of the boundary value problem (1), (2). The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34">View MathML</a> are eigenfunctions and there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M45">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M46">View MathML</a>

(6)

Denote the normalizing numbers by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M47">View MathML</a>

The following relation is valid:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M48">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M49">View MathML</a>. In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M34">View MathML</a> are solutions of the problem (1), (2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M52">View MathML</a>

Multiplying the equations by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M56">View MathML</a>, respectively, adding them together, integrating from 0 to π and using the condition (2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M57">View MathML</a>

is found. From (6) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M58">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M59">View MathML</a>

The following two theorems are obtained by Huseynov and Latifova in [27].

Theorem 1 (i) The boundary value problem (1), (2) has a countable set of simple eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M13">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M62">View MathML</a>

(8)

(ii) The eigen vector-functions of problem (1), (2) can be represented in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M63">View MathML</a>

(iii) The normalizing numbers of problem (1), (2) have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M64">View MathML</a>

(9)

Theorem 2 (i) The system of eigen vector-functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) of problem (1), (2) is complete in space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M67">View MathML</a>.

(ii) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M68">View MathML</a>be an absolutely continuous vector-function on the segment<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M69">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M70">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M71">View MathML</a>

(10)

moreover, the series converges uniformly with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M41">View MathML</a>.

(iii) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M73">View MathML</a>series (10) converges in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M67">View MathML</a>; moreover, the Parseval equality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M75">View MathML</a>

(11)

From [27], the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M76">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M77">View MathML</a> is a positive number and this inequality is valid in the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M78">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M79">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M80">View MathML</a>) are zeros of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M81">View MathML</a> and δ is a sufficiently small number.

In Section 3, the fundamental equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M82">View MathML</a>

is derived by using the method by Gelfand-Levitan-Marchenko, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M83">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M84">View MathML</a>

In Section 4, we show that the fundamental equation has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30">View MathML</a> and the boundary value problem (1), (2) can be uniquely determined from the spectral data. In Section 5, the result is obtained from Lemma 6 that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23">View MathML</a> defined by (3) satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M87">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M88">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30">View MathML</a> is the solution of the fundamental equation. In Lemma 7, using the fundamental equation, the Parseval equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M90">View MathML</a>

is found. We demonstrate by using Lemma 6, Lemma 9, and Lemma 10 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M91">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) are spectral data of the boundary value problem (1), (2). Then necessary and sufficient conditions for the solvability of problem (1), (2) are obtained in Theorem 11. Finally, we give an algorithm of the construction of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20">View MathML</a> by the spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M91">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>).

Note that throughout this paper, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M96">View MathML</a> denotes the transposed matrix of ϕ.

3 Fundamental equation

Theorem 3For each fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M97">View MathML</a>the kernel<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30">View MathML</a>from the representation (3) satisfies the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M99">View MathML</a>

(13)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M100">View MathML</a>

(14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M101">View MathML</a>

(15)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M102">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M103">View MathML</a>are, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2) when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M104">View MathML</a>.

Proof According to (3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M105">View MathML</a>

(16)

It follows from (3) and (16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M106">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M107">View MathML</a>

Using the last two equalities, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M108">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M109">View MathML</a>

(17)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M110">View MathML</a>

It is easily found by using (14) and (15) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M111">View MathML</a>

(18)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M112">View MathML</a>. Then according to the expansion formula (10) in Theorem 2, we obtain uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M41">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M114">View MathML</a>

(19)

From (18), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M115">View MathML</a>

(20)

It follows from (3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M116">View MathML</a>

(21)

Taking into account (21) and expansion formula (10) in Theorem 2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M117">View MathML</a>

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M118">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M119">View MathML</a>

(22)

Now, we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M120">View MathML</a>

(23)

Using (7) and the residue theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M121">View MathML</a>

(24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M122">View MathML</a> is oriented counter-clockwise, N is a sufficiently large number. Taking into account the asymptotic formulas as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M123">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M124">View MathML</a>

and the relations ([20], Lemma 1.3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M125">View MathML</a>

it follows from (12) and (24) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M126">View MathML</a>

(25)

Thus, using (17), (19), (20), (22) (23), and (25), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M127">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M68">View MathML</a> can be chosen arbitrarily,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M129">View MathML</a>

is obtained. □

4 Uniqueness

Lemma 4For each fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M97">View MathML</a>, (13) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M131">View MathML</a>.

Proof When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M132">View MathML</a>, (13) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M133">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M134">View MathML</a>

(26)

Now, we shall prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M135">View MathML</a> is invertible, i.e. has a bounded inverse in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136">View MathML</a>.

Consider the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M138">View MathML</a>. From this and (24),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M139">View MathML</a>

We show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M140">View MathML</a>

In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M141">View MathML</a>

Thus, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M135">View MathML</a> is invertible in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136">View MathML</a>. Therefore the fundamental equation (13) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M144">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M145">View MathML</a> is completely continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136">View MathML</a>. Then it is sufficient to prove that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M147">View MathML</a>

(27)

has only the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M148">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M149">View MathML</a> be a non-trivial solution of (27). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M150">View MathML</a>

It follows from (14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M151">View MathML</a>

Using (21), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M152">View MathML</a>

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M153">View MathML</a> into the last two integrals, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M154">View MathML</a>

(28)

Using the Parseval equality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M155">View MathML</a>

it follows from (28) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M156">View MathML</a>

Since the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M157">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M80">View MathML</a>) is complete in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M67">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M160">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M161">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M135">View MathML</a> invertible in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M164">View MathML</a> is obtained. □

Theorem 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M165">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M166">View MathML</a>be two boundary value problems and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M167">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M168">View MathML</a>

Proof According to (14) and (15), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M169">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M170">View MathML</a>. Then, from the fundamental equation (13), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M171">View MathML</a>. It follows from (4) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M172">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M173">View MathML</a>. □

5 Reconstruction by spectral data

Let the real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M175">View MathML</a>) of the form (8) and (9) be given. Using these numbers, we construct the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M177">View MathML</a> by (14) and (15) and determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30">View MathML</a> from the fundamental equation (13).

Now, let us construct the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M23">View MathML</a> by (3) and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20">View MathML</a> by (4). From [2], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M177">View MathML</a> have a derivative in both variables and these derivatives belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M183">View MathML</a>.

Lemma 6The following relations hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M184">View MathML</a>

(29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M185">View MathML</a>

(30)

Proof Differentiating to x and y, (13), respectively, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M186">View MathML</a>

(31)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M187">View MathML</a>

(32)

It follows from (14) and (15) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M188">View MathML</a>

(33)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M189">View MathML</a>

(34)

and using the fundamental equation (13), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M190">View MathML</a>

(35)

Multiplying (31) on the left by B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M191">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M192">View MathML</a>

(36)

and multiplying (32) on the right by B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M193">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M194">View MathML</a>

(37)

Adding (36) and (37) and using (34), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M195">View MathML</a>

(38)

From (33), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M196">View MathML</a>

(39)

Integrating by parts and from (35)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M197">View MathML</a>

(40)

is obtained. Substituting (40) into (38) and dividing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M198">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M199">View MathML</a>

(41)

Multiplying (13) on the left by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20">View MathML</a> in the form of (4) and adding to (41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M201">View MathML</a>

(42)

is obtained. Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M202">View MathML</a>

we can rewrite (42) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M203">View MathML</a>

(43)

According to Lemma 4, the homogeneous equation (43) has only the trivial solution, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M204">View MathML</a>

(44)

Differentiating (3) and multiplying on the left by B, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M205">View MathML</a>

(45)

On the other hand, multiplying (3) on the left by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M206">View MathML</a> and then integrating by parts and using (35), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M207">View MathML</a>

(46)

It follows from (45) and (46) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M208">View MathML</a>

Taking into account (4) and (44),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M209">View MathML</a>

is obtained. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M210">View MathML</a>, from (3) we get (30). □

Lemma 7For each function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M211">View MathML</a>, the following relation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M212">View MathML</a>

(47)

Proof It follows from (3) and (21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M213">View MathML</a>

(48)

Using the expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M214">View MathML</a>

the fundamental equation (13) is transformed into the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M215">View MathML</a>

(49)

From (48), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M216">View MathML</a>

(50)

and for the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M217">View MathML</a> we have the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M218">View MathML</a>

(51)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M219">View MathML</a>

and using (48) it is transformed into the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M220">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M221">View MathML</a>

(52)

Similarly, in view of (50), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M222">View MathML</a>

(53)

According to (52),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M223">View MathML</a>

It follows from (49) and (51) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M224">View MathML</a>

(54)

From (18) and the Parseval equality we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M225">View MathML</a>

Taking into account (54), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M226">View MathML</a>

whence, by (52) and (53),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M227">View MathML</a>

is obtained, i.e., (47) is valid. □

Corollary 8For any function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M229">View MathML</a>, the following relation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M230">View MathML</a>

(55)

Lemma 9The relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M231">View MathML</a>

(56)

is valid.

Proof (1) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M232">View MathML</a>. Consider the series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M233">View MathML</a>

(57)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M234">View MathML</a>

(58)

Using Lemma 6 and integrating by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M235">View MathML</a>

Applying the asymptotic formulas in Theorem 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M236">View MathML</a> is found. Consequently the series (57) converges absolutely and uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M69">View MathML</a>. According to (55) and (58), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M238">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M239">View MathML</a> is arbitrary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M240">View MathML</a> is obtained, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M241">View MathML</a>

(59)

(2) Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M242">View MathML</a> and assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M243">View MathML</a>. Then, by virtue of (59),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M244">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M245">View MathML</a>

The system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M246">View MathML</a> is minimal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M247">View MathML</a> and consequently by (3), the system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M248">View MathML</a> is minimal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M249">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M250">View MathML</a> and we obtain (56). □

Lemma 10For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M252">View MathML</a>

is valid.

Proof It is easily found that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M253">View MathML</a>

According to (56), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M254">View MathML</a>

(60)

We shall prove that for any n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M255">View MathML</a>. Assume the contrary, i.e. there exists m such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M256">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M257">View MathML</a>, it follows from (60) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M258">View MathML</a>. On the other hand, since as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M259">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M260">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M255">View MathML</a>. This contradicts the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M257">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M255">View MathML</a> for any n. From (60), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M265">View MathML</a>

Thus, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M266">View MathML</a>, for any n. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M267">View MathML</a>

we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M268">View MathML</a>, and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M269">View MathML</a> is obtained. □

Theorem 11For the sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M80">View MathML</a>) to be the spectral data for a certain boundary value problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M165">View MathML</a>of the form (1), (2) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M273">View MathML</a>, it is necessary and sufficient that the relations (8) and (9) hold.

Proof Necessity of the problem is proved in [27]. Let us prove the sufficiency. Let the real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) of the form (8) and (9) be given. It follows from Lemma 6, Lemma 9, and Lemma 10 that the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M276">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) are spectral data for the constructed boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M165">View MathML</a>. The theorem is proved. □

The algorithm of the construction of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20">View MathML</a> by the spectral data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M276">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) follows from the proof of the theorem:

(1) By the given numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M276">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M15">View MathML</a>) the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M177">View MathML</a> are constructed, respectively, by (14) and (15).

(2) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M30">View MathML</a> is found from (13).

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/110/mathml/M20">View MathML</a> is calculated by (4).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).

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