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On the spectral investigation of the scattering problem for some version of one-dimensional Schrödinger equation with turning point

Zaki FA El-Raheem1* and AH Nasser2

Author Affiliations

1 Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

2 Department of Mathematics, Faculty of Industrial Education, Helwan University, Cairo, Egypt

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Boundary Value Problems 2014, 2014:97  doi:10.1186/1687-2770-2014-97


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/97


Received:30 January 2014
Accepted:1 April 2014
Published:6 May 2014

© 2014 El-Raheem and Nasser; 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 we introduce and investigate the eigenvalues and the normalizing numbers as well as the scattering function for some version of the one-dimensional Schrödinger equation with turning point on the half line.

MSC: 58C40, 34L25.

Keywords:
initial value problem; the eigenvalues; normalizing numbers; scattering function; asymptotic formula

1 Introduction

The solution of many problems of mathematical physics are reduced to the spectral investigation of a differential operator. The differential operator is called regular if its domain is finite and its coefficients are continuous, otherwise it is called a singular differential operator. The Sturm-Liouville theory occupies a central position in the spectral theory of regular operator. During the development of quantum mechanics there was an increase in the interest of spectral theory of singular operators, on which we will restrict our attention. The first basic role in the development of the spectral theory of singular operators dates back to Titchmarsh [1]. He gave a new approach in the spectral theory of singular differential operator of the second order by using contour integration. Also Levitan [2] gave a new method, he obtained the eigenfunction expansion in an infinite interval by taking the limit of a regular case. In the last 35 or so years, due to the needs of mathematical physics, in particular, quantum mechanics, the question of solving various spectral problems with explosive factor has appeared in the study of geophysics and electromagnetic fields; see [3,4]. The spectral theory of differential operators with explosive factor is studied by Tikhonov [5], Gasymov [6]. For earlier results on various aspects of solvability theory of boundary value problems and spectral theory in the half line case, the situation closely related to the principal topic of this paper, we refer, for instance, to [7-10]. Notice that the paper [11] presented an approximate construction of the Jost function for some Sturm-Liouville boundary value problem in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M1">View MathML</a> by means of the collocation method. In the present paper we introduce and investigate the eigenvalues and the normalizing numbers as well as the scattering function for some version of the one-dimensional Schrödinger equation with turning point on the half line as in (1.1), (1.2). In [12,13], and [14] the weight functions introduced are considered as applications of the discontinuous wave speed problem on a non-homogeneous medium as in our case, while the introduction of the weight function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M2">View MathML</a> which is given by (1.3) as ± signs causes an excess of analytical difficulties. In [15] the author studied the spectral property in a finite interval, while in the present work we consider the half line which gives rise both to a continuous and a discrete spectrum; the latter is treated by the scattering function. In [16] the author considered the weight function of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M3">View MathML</a>

and the spectra were both continuous and discrete as in our problem. We must notice that the result of this paper is a starting point in calculating the regularized trace formula and solving the inverse scattering problem, which will be investigated later on.

Consider the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M4">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M5">View MathML</a>

(1.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M6">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M7">View MathML</a> is a finite real valued function which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M8">View MathML</a>

and μ is a complex spectral parameter. To study the eigenvalues of (1.1)-(1.2), we first consider the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M10">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M10">View MathML</a> problem (1.1)-(1.2) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M13">View MathML</a>

(1.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M14">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M15">View MathML</a>

(1.6)

From now on we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M16">View MathML</a> because according to (1.6) μ covers all the complex plane. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M17">View MathML</a> the solution of (1.4) with the initial conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M19">View MathML</a>. According to (1.3), (1.4) is equivalent to the two equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M20">View MathML</a>

(1.7)

It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M21">View MathML</a>

(1.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M23">View MathML</a> are calculated from the requirements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M25">View MathML</a>, so that (1.8) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M26">View MathML</a>

(1.9)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M27">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M28">View MathML</a> does not belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M29">View MathML</a> also, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M31">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M32">View MathML</a> whereas <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M33">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M32">View MathML</a>, so that it is convenient to consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M35">View MathML</a>

(1.10)

as the equation of the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M36">View MathML</a>.

From this we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M38">View MathML</a> or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M39">View MathML</a>

(1.11)

Together with the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M40">View MathML</a> of (1.4) we introduce the second solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M41">View MathML</a>, which is known as the Jost solution. This solution is defined by the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M42">View MathML</a>

(1.12)

With the aid of (1.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M43">View MathML</a>

where the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M45">View MathML</a> are calculated from the requirements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M47">View MathML</a>, and the solution becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M48">View MathML</a>

(1.13)

It should be noted, here, that the equation of the eigenvalues can be obtained, also, from the condition that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M49">View MathML</a>; this condition implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M50">View MathML</a>, which is the same as (1.10).

Now for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M52">View MathML</a> we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M53">View MathML</a> the solution of (1.1) which satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M54">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M55">View MathML</a>, (1.1) takes the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M56">View MathML</a>, and in the following, we study its solution and the related spectrum. From [4] this solution has the following representation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M57">View MathML</a>

(1.14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M60">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M61">View MathML</a>, the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62">View MathML</a> has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M63">View MathML</a>

(1.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M65">View MathML</a> is the fundamental system of solutions of (1.1) subject to the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M66">View MathML</a>

(1.16)

where the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M68">View MathML</a> are calculated from the requirements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M70">View MathML</a>, from which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M71">View MathML</a>

(1.17)

Further, (1.1), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M72">View MathML</a>, takes the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M73">View MathML</a>, and the fundamental system of solution of this follows from [[4], p.18] by the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M74">View MathML</a>

(1.18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M75">View MathML</a>

(1.19)

Now we find the characteristic equation of the eigenvalues of (1.1)-(1.2). Since the solution (1.15) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M77">View MathML</a> it follows that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M78">View MathML</a> to be an eigenvalue, it must satisfy the initial condition (1.2), namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M79">View MathML</a>

(1.20)

From (1.15) and (1.16) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M80">View MathML</a>

(1.21)

In the following lemmas we study some properties of the eigenvalues of problem (1.1)-(1.2).

Lemma 1.1Under the conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M81">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M82">View MathML</a>), the roots of (1.20), for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M77">View MathML</a>, are simple and lie only on the imaginary axis.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M84">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M85">View MathML</a>, be a zero of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M86">View MathML</a>, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M87">View MathML</a>

(1.22)

We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M89">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M90">View MathML</a> is a solution of (1.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M91">View MathML</a>

(1.23)

multiplying both sides of this by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M92">View MathML</a> and integrating both sides from 0 to ∞, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M93">View MathML</a>

Integrating the first integral by parts and using (1.22), (1.15) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M94">View MathML</a>

(1.24)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M95">View MathML</a>, from which we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M96">View MathML</a> is real and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M97">View MathML</a> is pure imaginary. We turn now to the proof that the roots are simple from (1.22), this is carried out by proving that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M98">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M99">View MathML</a>, where ‘dot’ denotes differentiation with respect to λ.

Integrating the difference <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M100">View MathML</a> with respect to x from 0 to ∞ and using (1.20) we get after some calculation that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M101">View MathML</a>

(1.25)

We prove the reality of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M104">View MathML</a> the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M105">View MathML</a> is real because reality of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M106">View MathML</a> comes from the reality of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M107">View MathML</a>.

To prove that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M108">View MathML</a>, we observe that φ and θ are real. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M109">View MathML</a>; since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M64">View MathML</a> is a solution of (1.1)-(1.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M111">View MathML</a>

(1.26)

Taking the conjugate of (1.26) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M112">View MathML</a>

(1.27)

It is clear, from (1.26) and (1.27), that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M113">View MathML</a>. In a similar way we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M114">View MathML</a> is also real so that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M116">View MathML</a> is real from which we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M117">View MathML</a> and (1.25) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M118">View MathML</a>

(1.28)

From (1.28) we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M119">View MathML</a>, which completes the proof. □

Remark 1 For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M121">View MathML</a>, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M122">View MathML</a> is the eigenfunction of problem (1.1)-(1.2) that corresponds to the negative eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M123">View MathML</a>.

Lemma 1.2For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124">View MathML</a>the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M125">View MathML</a>does not tend to zero, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M126">View MathML</a>

(1.29)

Proof Since the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62">View MathML</a> is the solution of (1.1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M128">View MathML</a> is also a solution, and it can be shown that these two solutions are linearly independent and their Wronskian is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M129">View MathML</a>

(1.30)

so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M130">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M131">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M133">View MathML</a> is a fundamental system of solutions of (1.1). In particular, putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M134">View MathML</a> into (1.30) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M135">View MathML</a>

(1.31)

To prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M138">View MathML</a>, assume to the contrary i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M138">View MathML</a>. From (1.31) and (1.20) we reach to contradiction to the assumption, and, consequently, we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M138">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M145">View MathML</a>. □

Lemma 1.3For all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M124">View MathML</a>the following equality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M147">View MathML</a>

(1.32)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M148">View MathML</a>is the solution of problem (1.1)-(1.2) and the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M149">View MathML</a>

(1.33)

satisfies the properties

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M150">View MathML</a>

(1.34)

It should be noted here that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M151">View MathML</a> defined by (1.33) is called the scattering function of problem (1.1)-(1.2) and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M152">View MathML</a> is called the denominator of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M153">View MathML</a>.

Proof As mentioned before (1.30) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M156">View MathML</a> is a fundamental system of solutions of (1.1)-(1.2), so that any linear combination of them is again a solution of (1.1)-(1.2):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M157">View MathML</a>

(1.35)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M159">View MathML</a> are calculated from the initial conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M161">View MathML</a> in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M162">View MathML</a>

(1.36)

Substituting (1.36) into (1.35) we arrive at the required formula (1.32). Further, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M163">View MathML</a>, it follows from (1.33) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M164">View MathML</a>

from which we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M165">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M166">View MathML</a>

 □

2 The asymptotic formulas of eigenvalues and normalizing numbers

The eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M167">View MathML</a> of problem (1.1)-(1.2) are the roots of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M168">View MathML</a>

(2.1)

In the following we prove that (2.1) has an infinite number of roots and find their asymptotic formula. From (1.15), (1.17), (1.18), and (1.19) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M169">View MathML</a>

(2.2)

Now, we calculate the asymptotic formula of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M171">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M173">View MathML</a>. Integrating (1.15) by parts we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M77">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M176">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M177">View MathML</a>

(2.4)

Similarly from (1.18) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M178">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M179">View MathML</a>

(2.6)

The following group of inequalities follows from (2.3)-(2.6):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M180">View MathML</a>

(2.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M181">View MathML</a>

(2.8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M182">View MathML</a>

(2.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M183">View MathML</a>

(2.10)

Substituting (2.7)-(2.10) into (2.2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M184">View MathML</a>

(2.11)

comparing (1.10) and (2.11) we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M186">View MathML</a> have the same number of zeros inside the quadratic contour <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M187">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M188">View MathML</a>, but since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M189">View MathML</a> has exactly n zeros, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M191">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M192">View MathML</a> has an infinite number of zeros, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M193">View MathML</a>, with limiting point at infinity. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M194">View MathML</a> the zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M195">View MathML</a>, so that, by the Rouche theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M196">View MathML</a>

(2.12)

To make (2.12) more accurate, we must refine (2.11). With the aid of Lemma 1.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M194">View MathML</a> lies on the imaginary axis, so that it is sufficient to know the asymptotic of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M198">View MathML</a> for small λ. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M200">View MathML</a>, we find the asymptotic formula of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M201">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M202">View MathML</a>. From (2.3), (2.4), (2.5), and (2.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M203">View MathML</a>

(2.13)

substituting (2.13) into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M204">View MathML</a>, and putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M205">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M206">View MathML</a>

(2.14)

and from this and by virtue of the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M207">View MathML</a>n, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M208">View MathML</a>

(2.15)

From (2.12), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M209">View MathML</a>

(2.16)

The estimation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M210">View MathML</a> follows from (2.15) and (2.16) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M211">View MathML</a>

(2.17)

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M212">View MathML</a>

(2.18)

Finally

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M213">View MathML</a>

(2.19)

Definition (The normalizing numbers)

The numbers

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M214">View MathML</a>

(2.20)

are called the normalizing numbers of problem (1.1)-(1.2) (notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M215">View MathML</a> are the eigenfunctions of problem (1.1)-(1.2) corresponding to the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M216">View MathML</a>). From (1.28) and the reality of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M217">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M218">View MathML</a>

(2.21)

To evaluate the asymptotic formula of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M219">View MathML</a> we evaluate the asymptotic formula of the right hand side of (2.21). From (1.15), (1.17) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M220">View MathML</a>

(2.22)

where dots and dashes denote the differentiation with respect to λ and x, respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M67">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M222">View MathML</a> are given by (1.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M223">View MathML</a>

from which it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M224">View MathML</a>

(2.23)

From (1.18), using integration by parts and then putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M225">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M109">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M227">View MathML</a>

(2.24)

From (1.19), carrying out a similar calculation with respect to θ, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M228">View MathML</a>

(2.25)

With the aid of (1.15), similar expressions can be calculated with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M229">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M230">View MathML</a>

(2.26)

From (2.21) and (2.22), the normalizing numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M219">View MathML</a> can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M232">View MathML</a>

(2.27)

We substitute (2.23), (2.24), (2.25), and (2.26) into (2.27), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M233">View MathML</a>, and we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M234">View MathML</a>

(2.28)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M237">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M238">View MathML</a>. Further, from (2.16) and (2.17) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M239">View MathML</a>

(2.29)

By substituting from (2.29) into (2.28) we obtain the required asymptotic formula for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M219">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M241">View MathML</a>

(2.30)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/97/mathml/M242">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The two authors typed read and approved the final manuscript also they contributed to each part of this work equally.

Acknowledgements

We are indebted to an anonymous referee for a detailed reading of the manuscript and useful comments and suggestions, which helped us improve this work. This work is supported by the Research Support Unit of Alexandria University.

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