SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Inverse eigenvalue problems for a discontinuous Sturm-Liouville operator with two discontinuities

Yalçın Güldü

Author Affiliations

Department of Mathematics, Faculty of Sciences, Cumhuriyet University, Sivas, 58140, Turkey

Boundary Value Problems 2013, 2013:209  doi:10.1186/1687-2770-2013-209

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/209


Received:18 June 2013
Accepted:26 August 2013
Published:11 September 2013

© 2013 Güldü; 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 cited.

Abstract

In this paper, we consider a discontinuous Sturm-Liouville operator with parameter-dependent boundary conditions and two interior discontinuities. We obtain eigenvalues and eigenfunctions together with their asymptotic approximate formulas. Then, we give some uniqueness theorems by using Weyl function and spectral data, which are called eigenvalues and normalizing constants for solution of inverse problem.

MSC: 34A55, 34B24, 34L05.

Keywords:
Sturm-Liouville problem; eigenvalues; eigenfunctions; transmission conditions; Weyl function

1 Introduction

It is well known that the theory of Sturm-Liouville problems is one of the most actual and extensively developing fields of theoretical and applied mathematics, since it is an important tool in solving many problems in mathematical physics (see [1-4]). In recent years, there has been increasing interest in spectral analysis of discontinuous Sturm-Liouville problems with eigenvalue-linearly and nonlinearly dependent boundary conditions [1,5-12]. Various physics applications of such problems can be found in [1,3,4,13-19] and corresponding bibliography cited therein.

Some boundary value problems with discontinuity conditions arise in heat and mass transfer problems, mechanics, electronics, geophysics and other natural sciences (see [3] also [20-29]). For instance, discontinuous inverse problems appear in electronics for building parameters of heterogeneous electronic lines with attractive technical characteristics [20,30,31]. Such discontinuity problems also appear in geophysical forms for oscillations of the earth [32,33]. Furthermore, discontinuous inverse problems appear in mathematics for exploring spectral properties of some classes of differential and integral operators.

Inverse problems of spectral analysis form recovering operators by their spectral data. The inverse problem for the classical Sturm-Liouville operator was studied first by Ambarsumian in 1929 [34] and then by Borg in 1945 [35]. After that, direct and inverse problems for Sturm-Liouville operator have been extended to so many different areas.

We consider a discontinuous Sturm-Liouville problem L with function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M1">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M2">View MathML</a>

(1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M3">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M5">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M6">View MathML</a> are given positive real numbers; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M7">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M8">View MathML</a> is a complex spectral parameter; boundary conditions at the endpoints

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M9">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M10">View MathML</a>

(3)

with discontinuity conditions at two points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M12">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M13">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M14">View MathML</a>

(5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M15">View MathML</a>

(6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M16">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M20">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22">View MathML</a>) are real numbers and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M23">View MathML</a>

In the present paper, we construct a linear operator T in a suitable Hilbert space such that problem (1)-(7) and the eigenvalue problem for operator T coincide. We investigate eigenvalues and eigenfunctions together with their asymptotic behaviors of operator T. Besides, we study some uniqueness theorems according to Weyl function and spectral data, which are called eigenvalues and normalizing constants.

2 Operator formulation and spectral properties

We make known the inner product in the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M24">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M25">View MathML</a>, ℂ denotes the Hilbert space of complex numbers and a self-adjoint operator T defined on H such that (1)-(7) can be dealt with as the eigenvalue problem of operator T. We define an inner product in H by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M26">View MathML</a>

(8)

for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M27">View MathML</a>

Consider the operator T defined by the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M28">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M29">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M30">View MathML</a> and also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M31">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M32">View MathML</a> are satisfied for f.

Thus, we can rewrite the considered problem (1)-(7) in the operator form as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M33">View MathML</a>.

Theorem 1The operatorTis symmetric inH.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M34">View MathML</a>. By two partial integrations, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M35">View MathML</a>

where by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M36">View MathML</a>, we denote the Wronskian of the functions f and g as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M37">View MathML</a>

Since f and g satisfy the boundary conditions (2)-(3) and transmission conditions (4)-(7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M38">View MathML</a>

Thus, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M39">View MathML</a>, i.e., T is symmetric. □

Lemma 1Problem (1)-(7) can be considered as the eigenvalue problem of the symmetric operatorT.

Corollary 1All eigenvalues and eigenfunctions of problem (1)-(7) are real, and two eigenfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M40">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M41">View MathML</a>, corresponding to different eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M42">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M43">View MathML</a>, are orthogonal in the sense of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M44">View MathML</a>

We define the solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M45">View MathML</a>

of equation (1) by the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M46">View MathML</a>

(9)

and similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M47">View MathML</a>

(10)

respectively.

These solutions are entire functions ofλfor each fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M48">View MathML</a>and satisfy the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M49">View MathML</a>

for each eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M50">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M51">View MathML</a>

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M53">View MathML</a>.

Then the following integral equations and also asymptotic behaviors hold for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M54">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M55">View MathML</a>

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M53">View MathML</a>.

Then the following integral equations and also asymptotic behaviors hold for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M54">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M59">View MathML</a>

The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60">View MathML</a>is called the characteristic function, and numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M61">View MathML</a>are called the normalizing constants of problem (1)-(7) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M62">View MathML</a>

(11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M63">View MathML</a>

(12)

Lemma 4The following equality holds for each eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M50">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M65">View MathML</a>

Proof Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M66">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M67">View MathML</a>

After that, add and subtract <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60">View MathML</a> on the left-hand side of the last equality, and by using conditions (2)-(7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M69">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M70">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M72">View MathML</a> is obtained by using the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M73">View MathML</a>

and (12). □

Corollary 2The eigenvalues of problemLare simple.

Lemma 5[36]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M74">View MathML</a>be the set of real numbers satisfying the inequalities<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M75">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M76">View MathML</a>be the set of complex numbers. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M77">View MathML</a>, then the roots of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M78">View MathML</a>

have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M79">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M80">View MathML</a>is a bounded sequence.

Now, from Lemma 2 and (11), we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M81">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M82">View MathML</a>.

We can see that non-zero roots, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M83">View MathML</a> of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M84">View MathML</a>, are real and analytically simple.

Furthermore, it can be proved by using Lemma 5 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M85">View MathML</a>

(13)

Theorem 2The eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M86">View MathML</a>have the following asymptotic behavior for sufficiently largen:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M87">View MathML</a>

(14)

Proof Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M88">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M89">View MathML</a> and δ is a sufficiently small number. The relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M90">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M91">View MathML</a>

are valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M92">View MathML</a>.

Then, by Rouche’s theorem that the number of zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M93">View MathML</a> coincides with the number of zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M95">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M96">View MathML</a> zeros, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M97">View MathML</a>. In the annulus, between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60">View MathML</a> has accurately one positive zero, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M101">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M102">View MathML</a>. So, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M103">View MathML</a>. Applying to Rouche’s theorem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M104">View MathML</a> for sufficiently small ε and sufficiently large n, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M105">View MathML</a>. Finally, we obtain the asymptotic formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M106">View MathML</a>

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M107">View MathML</a>

which are independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M108">View MathML</a> and are entire functions such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M111">View MathML</a>.

It can be easily seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M112">View MathML</a>

 □

Example Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M119">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M127">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M129">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M130">View MathML</a>

the eigenvalues of the boundary value problem (1)-(7) satisfy the following asymptotic formulae:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M131">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M132">View MathML</a>.

3 Inverse problems

In this section, we study the inverse problems for the reconstruction of the boundary value problem (1)-(7) by Weyl function and spectral data.

We consider the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M133">View MathML</a> with the same form of L but with different coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22">View MathML</a>.

If a certain symbol α denotes an object related to L, then the symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M142">View MathML</a> denotes the corresponding object related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M133">View MathML</a>.

The Weyl function Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M144">View MathML</a> be a solution of equation (1), which satisfies the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M145">View MathML</a> and transmissions (4)-(7).

Assume that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M146">View MathML</a> is the solution of equation (1) that satisfies the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M148">View MathML</a> and the transmission conditions (4)-(7).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M149">View MathML</a>, the functions χ and φ are linearly independent. Therefore, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M150">View MathML</a> can be represented by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M151">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M152">View MathML</a>

(15)

that is called the Weyl solution, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M153">View MathML</a>

(16)

is called the Weyl function.

Theorem 3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M154">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M155">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M156">View MathML</a>, a.e. and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22">View MathML</a>.

Proof We introduce a matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M164">View MathML</a> by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M165">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M166">View MathML</a>

(17)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M167">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M168">View MathML</a>

Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M169">View MathML</a>, then the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M171">View MathML</a> are entire in λ for each fixed x.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M173">View MathML</a>, where w is sufficiently small number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M175">View MathML</a> are square roots of the eigenvalues of the problem L and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M133">View MathML</a>, respectively. It is easily shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M177">View MathML</a>

(18)

are valid for sufficiently large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M178">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M181">View MathML</a>. Hence, Lemma 2 and (18) yield that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M182">View MathML</a>

(19)

According to (19), and Liouville’s theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M184">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M185">View MathML</a>. By virtue of (17), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M186">View MathML</a>

(20)

It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M187">View MathML</a>

and similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M188">View MathML</a>. Thus, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M189">View MathML</a>.

Otherwise, the following asymptotic expressions hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M190">View MathML</a>

(21)

Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M192">View MathML</a>. From (20)-(21), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M193">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M194">View MathML</a> and also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M195">View MathML</a>

(22)

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M196">View MathML</a> in (22), we contradict <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M197">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M199">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M202">View MathML</a>. Hence, from equation (1) and transmission conditions (4)-(7), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M156">View MathML</a>, a.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M206">View MathML</a>, and from (9) and (10), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M210">View MathML</a>. □

Lemma 6The following representation holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M211">View MathML</a>

Proof Weyl function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M212">View MathML</a> is a meromorphic function with respect to λ, which has simple poles at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M50">View MathML</a>. Therefore, we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M214">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M215">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M216">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M217">View MathML</a>

(23)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M218">View MathML</a>, where ε is a sufficiently small number. Consider the contour integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M219">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M220">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M221">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M222">View MathML</a>

satisfies. Using this equality and (16), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M223">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M224">View MathML</a>. As a result, the residue theorem and (23) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M225">View MathML</a>

 □

Theorem 4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M226">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M227">View MathML</a>for alln, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M228">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M229">View MathML</a>, a.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22">View MathML</a>. Hence, problem (1)-(7) is uniquely determined by spectral data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M237">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M226">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M239">View MathML</a> for all n, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M154">View MathML</a> by Lemma 6. Therefore, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M155">View MathML</a> by Theorem 3.

Let us consider the boundary value problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M242">View MathML</a> that we get the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M243">View MathML</a> instead of condition (2) in L. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M244">View MathML</a> be the eigenvalues of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M242">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M246">View MathML</a> are zeros of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M247">View MathML</a>

 □

Theorem 5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M226">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M249">View MathML</a>for alln, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M252">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M22">View MathML</a>.

Hence, the problemLis uniquely determined by the sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M254">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M255">View MathML</a>, except coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M256">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M19">View MathML</a>.

Proof Since the characteristic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M259">View MathML</a> are entire of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M260">View MathML</a>, functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M259">View MathML</a> are uniquely determined up to multiplicative constant with their zeros by Hadamard’s factorization theorem [37]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M263">View MathML</a>

where C and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M264">View MathML</a> are constants dependent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M255">View MathML</a>, respectively. Therefore, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M267">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M249">View MathML</a> for all n, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M269">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M270">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M271">View MathML</a>. As a result, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/209/mathml/M154">View MathML</a> by (16). So, the proof is completed by Theorem 3. □

Competing interests

The author declares that he has no competing interests.

References

  1. Fulton, CT: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. A. 77, 293–308 (1977). Publisher Full Text OpenURL

  2. Kobayashi, M: Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials. Comput. Math. Appl.. 18(4), 355–364 (1989)

  3. Likov, AV, Mikhailov, YA: The Theory of Heat and Mass Transfer. Qosenergaizdat (1963)

  4. Shkalikov, AA: Boundary value problems for ordinary differential equations with a parameter in boundary conditions. Tr. Semin. Im. I.G. Petrovskogo. 9, 190–229 (1983)

  5. Binding, PA, Browne, PJ, Seddighi, K: Sturm-Liouville problems with eigenparameter dependent boundary conditions. Proc. Edinb. Math. Soc.. 37, 57–72 (1993)

  6. Binding, PA, Browne, PJ, Watson, BA: Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. J. Lond. Math. Soc.. 62, 161–182 (2000). Publisher Full Text OpenURL

  7. Binding, PA, Browne, PJ, Watson, BA: Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter. J. Math. Anal. Appl.. 291, 246–261 (2004)

  8. Fulton, CT: Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. A. 87, 1–34 (1980). Publisher Full Text OpenURL

  9. Mennicken, R, Schmid, H, Shkalikov, AA: On the eigenvalue accumulation of Sturm-Liouville problems depending nonlinearly on the spectral parameter. Math. Nachr.. 189, 157–170 (1998). Publisher Full Text OpenURL

  10. Russakovskii, EM: Operator treatment of boundary problems with spectral parameters entering via polynomials in the boundary conditions. Funct. Anal. Appl.. 9, 358–359 (1975)

  11. Schmid, H, Tretter, C: Singular Dirac systems and Sturm-Liouville problems nonlinear in the spectral parameter. J. Differ. Equ.. 181(2), 511–542 (2002). Publisher Full Text OpenURL

  12. Ozkan, AS, Keskin, B: Spectral problems for Sturm-Liouville operator with boundary and jump conditions linearly dependent on the eigenparameter. Inverse Probl. Sci. Eng.. 20, 799–808 (2012). Publisher Full Text OpenURL

  13. Akdoğan, Z, Demirci, M, Mukhtarov, OS: Sturm-Liouville problems with eigendependent boundary and transmissions conditions. Acta Math. Sci.. 25(4), 731–740 (2005)

  14. Akdoğan, Z, Demirci, M, Mukhtarov, OS: Discontinuous Sturm-Liouville problem with eigenparameter-dependent boundary and transmission conditions. Acta Appl. Math.. 86, 329–334 (2005). Publisher Full Text OpenURL

  15. Binding, PA, Browne, PJ, Watson, BA: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter II. J. Comput. Appl. Math.. 148, 147–169 (2002). Publisher Full Text OpenURL

  16. Kerimov, NB, Memedov, KK: On a boundary value problem with a spectral parameter in the boundary conditions. Sib. Mat. Zh.. 40(2), 325–335 (1999) (English translation: Sib. Math. J. 40(2), 281-290 (1999))

  17. Mukhtarov, OS, Kadakal, M, Muhtarov, FS: On discontinuous Sturm-Liouville problem with transmission conditions. J. Math. Kyoto Univ.. 444, 779–798 (2004)

  18. Tunç, E, Muhtarov, OS: Fundamental solution and eigenvalues of one boundary value problem with transmission conditions. Appl. Math. Comput.. 157, 347–355 (2004). Publisher Full Text OpenURL

  19. Yakubov, S: Completeness of Root Functions of Regular Differential Operators, Longman, New York (1994)

  20. Meschanov, VP, Feldstein, AL: Automatic Design of Directional Couplers, Sviaz, Moscow (1980)

  21. Tikhonov, AN, Samarskii, AA: Equations of Mathematical Physics, Pergamon, Oxford (1990)

  22. Voitovich, NN, Katsenelbaum, BZ, Sivov, AN: Generalized Method of Eigen-Vibration in the Theory of Diffraction, Nauka, Moscow (1997)

  23. McNabb, A, Anderssen, R, Lapwood, E: Asymptotic behavior of the eigenvalues of a Sturm-Liouville system with discontinuous coefficients. J. Math. Anal. Appl.. 54, 741–751 (1976). Publisher Full Text OpenURL

  24. Shieh, CT, Yurko, VA: Inverse nodal and inverse spectral problems for discontinuous boundary value problems. J. Math. Anal. Appl.. 347, 266–272 (2008). Publisher Full Text OpenURL

  25. Willis, C: Inverse Sturm-Liouville problems with two discontinuities. Inverse Probl.. 1, 263–289 (1985). Publisher Full Text OpenURL

  26. Yang, CF: An interior inverse problem for discontinuous boundary-value problems. Integral Equ. Oper. Theory. 65, 593–604 (2009). Publisher Full Text OpenURL

  27. Yang, CF, Yang, XP: An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions. Appl. Math. Lett.. 22, 1315–1319 (2009). Publisher Full Text OpenURL

  28. Yang, Q, Wang, W: Asymptotic behavior of a differential operator with discontinuities at two points. Math. Methods Appl. Sci.. 34, 373–383 (2011)

  29. Yurko, VA: Integral transforms connected with discontinuous boundary value problems. Integral Transforms Spec. Funct.. 10, 141–164 (2000). Publisher Full Text OpenURL

  30. Litvinenko, ON, Soshnikov, VI: The Theory of Heterogenious Lines and Their Applications in Radio Engineering. Moscow (1964)

  31. Shepelsky, DG: The inverse problem of reconstruction of the medium’s conductivity in a class of discontinuous and increasing functions. Spectral Operator Theory and Related Topics, pp. 209–232. Am. Math. Soc., Providence (1994)

  32. Anderssen, RS: The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of torsional eigenfrequencies of the earth. Geophys. J. R. Astron. Soc.. 50, 303–309 (1997)

  33. Lapwood, FR, Usami, T: Free Oscillations of the Earth, Cambridge University Press, Cambridge (1981)

  34. Ambarsumian, VA: Über eine frage der eigenwerttheorie. Z. Phys.. 53, 690–695 (1929). Publisher Full Text OpenURL

  35. Borg, G: Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Bestimmung der differentialgleichung durch die eigenwerte. Acta Math.. 78, 1–96 (1946). Publisher Full Text OpenURL

  36. Zhdanovich, VF: Formulae for the zeros of Dirichlet polynomials and quasi-polynomials. Dokl. Akad. Nauk SSSR. 135(8), 1046–1049 (1960)

  37. Titchmarsh, EC: The Theory of Functions, Oxford University Press, London (1939)