This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Spectral asymptotics of self-adjoint fourth order boundary value problems with eigenvalue parameter dependent boundary conditions

Manfred Möller* and Bertin Zinsou

Author Affiliations

The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

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Boundary Value Problems 2012, 2012:106  doi:10.1186/1687-2770-2012-106


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/106


Received:8 May 2012
Accepted:17 September 2012
Published:4 October 2012

© 2012 Möller and Zinsou; 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

A regular fourth order differential equation with λ-dependent boundary conditions is considered. For four distinct cases with exactly one λ-independent boundary condition, the asymptotic eigenvalue distribution is presented.

MSC: 34L20, 34B07, 34B08, 34B09.

Keywords:
fourth order boundary value problems; self-adjoint; boundary conditions; eigenvalue distribution; pure imaginary eigenvalues; spectral asymptotics

1 Introduction

Sturm-Liouville problems have attracted extensive attention due to their intrinsic mathematical challenges and their applications in physics and engineering. However, apart from classical Sturm-Liouville problems, also higher order ordinary linear differential equations occur in applications, with or without the eigenvalue parameter in the boundary conditions. Such problems are realized as operator polynomials, also called operator pencils. Some recent developments of higher order differential operators whose boundary conditions depend on the eigenvalue parameter, including spectral asymptotics and basis properties, have been investigated in [1-4]. General characterizations of self-adjoint boundary conditions have been presented in [5,6] for singular and (quasi-)regular problems. In all these cases, the minimal operator associated with an nth order differential equation must be symmetric, see [7,8] for necessary and sufficient conditions. A more general discussion on the spectra of fourth order differential operators can be found in [9,10].

The generalized Regge problem is realized by a second order differential operator which depends quadratically on the eigenvalue parameter and which has eigenvalue parameter dependent boundary conditions, see [11]. The particular feature of this problem is that the coefficient operators of this pencil are self-adjoint, and it is shown in [11] that this gives some a priori knowledge about the location of the spectrum. In [12] this approach has been extended to a fourth order differential equation describing small transversal vibrations of a homogeneous beam compressed or stretched by a force g. Separation of variables leads to a fourth order boundary problem with eigenvalue parameter dependent boundary conditions, where the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M1">View MathML</a>

depends quadratically on the eigenvalue parameter. This problem is represented by a quadratic operator pencil, in a suitably chosen Hilbert space, whose coefficient operators are self-adjoint. In [13] we have investigated a class of boundary conditions for which necessary and sufficient conditions were obtained such that the associated operator pencil consists of self-adjoint operators, while in [14] we have continued the work of [13] in the direction of [12] to derive eigenvalue asymptotics associated with boundary conditions which lead to self-adjoint operator representations. We have considered the particular case of boundary conditions which do not depend on the eigenvalue parameter at the left endpoint and depend on the eigenvalue parameter at the right endpoint.

In this paper, we extend the work of [14] to a class of boundary conditions where exactly one of the left endpoint boundary conditions does not depend on the eigenvalue parameter, while the remaining boundary conditions depend on the eigenvalue parameter.

We define the operator pencil in Section 2 and we discuss which boundary conditions are considered. In Section 3, the eigenvalue asymptotics for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a> are derived. In Section 4, it is shown that the boundary value problems under consideration are Birkhoff regular, which implies that the eigenvalues for general g are small perturbations of the eigenvalues for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a>. Hence, in Section 5, the first four terms of the eigenvalue asymptotics are found and are compared to those obtained in [14].

2 The quadratic operator pencil L

On the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M4">View MathML</a>, we consider the boundary value problem

(2.1)

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M8">View MathML</a> is a real valued function and (2.2) are separated boundary conditions where the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M9">View MathML</a> are constant or depend on λ linearly. The boundary conditions (2.2) are taken at the endpoint 0 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M10">View MathML</a> and at the endpoint a for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M11">View MathML</a>. Further, we assume for simplicity that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M12">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M13">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M14">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M16">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M19">View MathML</a>. We recall that the quasi-derivatives associated with (2.1) are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M20">View MathML</a>

see [[8], p.26].

Define

Assumption 2.1The numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M24">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M25">View MathML</a>are distinct as well as the numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M24">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M29">View MathML</a>.

We denote by U the collection of the boundary conditions (2.2) and define the following operators related to U:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M30">View MathML</a>

(2.3)

We put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M31">View MathML</a> and consider the linear operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M32">View MathML</a>, K and M in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M33">View MathML</a> with domains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M34">View MathML</a>

given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M35">View MathML</a>

It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M39">View MathML</a>. We associate a quadratic operator pencil

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M40">View MathML</a>

(2.4)

in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M33">View MathML</a> with the problem (2.1), (2.2).

The conditions under which the differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M32">View MathML</a> is self-adjoint are given in

Theorem 2.2 ([13], Theorem 1.2)

Denote by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M43">View MathML</a>the set ofpin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M44">View MathML</a>for theλ-independent boundary conditions and by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M45">View MathML</a>the corresponding set for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M46">View MathML</a>. Then the differential operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M32">View MathML</a>associated with this boundary value problem is self-adjoint if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M48">View MathML</a>for all boundary conditions of the form<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M49">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M50">View MathML</a>ifqis even in case<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M51">View MathML</a>or odd in case<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M53">View MathML</a>otherwise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M57">View MathML</a>.

Proposition 2.3The operator pencil<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M58">View MathML</a>is a Fredholm valued operator function with index 0. The spectrum of the Fredholm operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M58">View MathML</a>consists of discrete eigenvalues of finite multiplicities, and all eigenvalues of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M61">View MathML</a>, lie in the closed upper half-plane and on the imaginary axis and are symmetric with respect to the imaginary axis.

Proof As in [[12], Section 3], we can argue that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M63">View MathML</a> is a relatively compact perturbation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M64">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M64">View MathML</a> is well known to be a Fredholm operator. The statement on the location of the spectrum now follows as in [[12], Lemma 3.1]. □

We now consider the particular cases that exactly one of the boundary conditions at 0 depends on λ, whereas both boundary conditions at a depend on λ. Therefore, taking Assumption 2.1 and Theorem 2.2 into account, we have the four boundary conditions

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M70">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M71">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M72">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M73">View MathML</a>. Thus, we have 8 and 4 possible sets of boundary conditions at the endpoint 0 and a, respectively. Whence there are 32 different sets of boundary conditions. Recall that the parameter λ emanates from derivatives with respect to the time variable in the original partial differential equation, and it is reasonable that the highest space derivative occurs in the term without time derivative. Thus, the most relevant boundary conditions would have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M76">View MathML</a>. This leaves us with four different cases for the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M77">View MathML</a>.

These four cases are uniquely determined by the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M22">View MathML</a>, so that we will consider

Case 1: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M79">View MathML</a>; Case 2: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M80">View MathML</a>; Case 3: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M81">View MathML</a>; Case 4: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M82">View MathML</a>.

The corresponding boundary operators are then

(2.5)

(2.6)

(2.7)

(2.8)

3 Asymptotics of eigenvalues for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a>

In this section, we consider the boundary value problem (2.1), (2.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a>. We count all eigenvalues with their proper multiplicities and develop a formula for the asymptotic distribution of the eigenvalues, which is used to obtain the corresponding formula for general g. Observe that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a>, the quasi-derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M90">View MathML</a> coincide with the standard derivatives <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M91">View MathML</a>. We take the canonical fundamental system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M93">View MathML</a>, of (2.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M94">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M95">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M96">View MathML</a>. It is well known that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M97">View MathML</a> are analytic on ℂ with respect to λ. Putting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M98">View MathML</a>

the eigenvalues of the boundary value problem (2.1), (2.2) are the eigenvalues of the analytic matrix function M, where the corresponding geometric and algebraic multiplicities coincide, see [[15], Theorem 6.3.2].

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M99">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M100">View MathML</a>

it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M101">View MathML</a>

The second row of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M102">View MathML</a> has exactly two non-zero entries (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M103">View MathML</a>), and these non-zero entries are:

In Cases 1 and 2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M105">View MathML</a>;

In Cases 3 and 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M106">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M107">View MathML</a>.

Since the first row of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M102">View MathML</a> has exactly one entry 1 and all other entries zero, an expansion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M102">View MathML</a> with respect to the second row shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M110">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M111">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M112">View MathML</a>

In view of (2.7), (2.8) this gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M113">View MathML</a>

Each of the summands in ϕ is a product of a power in μ and a product of two sums of a trigonometric and a hyperbolic functions. The terms with the highest μ-powers in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M114">View MathML</a> are non-zero constant multiples of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M115">View MathML</a>

For the above four cases, we obtain:

We next give the asymptotic distributions of the zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M117">View MathML</a> with proper counting.

Lemma 3.1 Case 1: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a>has a zero of multiplicity 8 at 0, simple zeros at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M119">View MathML</a>

simple zeros at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M123">View MathML</a>, and no other zeros.

Case 2: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a>has a zero of multiplicity 4 at 0, exactly one simple zero<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125">View MathML</a>in each interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M126">View MathML</a>for positive integerskwith asymptotics

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M127">View MathML</a>

simple zeros at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M131">View MathML</a> , and no other zeros.

Case 3: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a>has a zero of multiplicity 6 at 0, simple zeros at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M133">View MathML</a>

simple zeros at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M137">View MathML</a>, and no other zeros.

Case 4: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a>has a zero of multiplicity 6 at 0, exactly one simple zero<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125">View MathML</a>in each interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M140">View MathML</a>for positive integerskwith asymptotics

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M141">View MathML</a>

simple zeros at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M121">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M122">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M137">View MathML</a>, and no other zeros.

Proof The result is obvious in Cases 1 and 3. Cases 2 and 4 only differ in the factor with the power of μ, and the multiplicity of the corresponding zero of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> at 0 is easy to verify. The choice of the indexing for the non-zero zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> in each case will become apparent later.

It, therefore, remains to describe the behavior of the non-zero zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> in Case 2. First, we are going to find the zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> on the positive real axis. One can observe that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M151">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M153">View MathML</a>, whence the positive zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> are those <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M155">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M156">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M158">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M159">View MathML</a> where the functions are defined, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M160">View MathML</a> is increasing with a positive derivative on each interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a>. On each of these intervals, the function moves from −∞ to ∞, thus we have exactly one simple zero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M164">View MathML</a> in each interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M126">View MathML</a>, where k is a positive integer, and no zero in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M166">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M167">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M168">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M169">View MathML</a>

The location of the zeros on the other three half-axes follows by repeated application of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M170">View MathML</a>.

The proof will be complete if we show that all zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> lie on the real or the imaginary axis. To this end, we observe that the product-to-sum formula for trigonometric functions gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M172">View MathML</a>

(3.1)

Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M174">View MathML</a>, it follows for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M175">View MathML</a> that

(3.2)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M177">View MathML</a> has a positive derivative on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M178">View MathML</a>, this function is strictly increasing, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M151">View MathML</a> therefore, implies by (3.2) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M180">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M181">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M182">View MathML</a>

is either real or pure imaginary. □

Proposition 3.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a>, there exists a positive integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M184">View MathML</a>such that the eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185">View MathML</a>, counted with multiplicity, of the problem (2.1), (2.5)-(2.8), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M186">View MathML</a>in Cases 1 and 2 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a>in Cases 3 and 4, can be enumerated in such a way that the eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185">View MathML</a>are pure imaginary for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M189">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M190">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M191">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M192">View MathML</a>, we can write<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M193">View MathML</a>, where the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M194">View MathML</a>have the following asymptotic representation as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195">View MathML</a>:

In particular, the number of pure imaginary eigenvalues is even in Cases 1 and 2 and odd in Cases 3 and 4.

Proof Case 4: A straightforward calculation gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M197">View MathML</a>

(3.3)

Up to the constant factor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M198">View MathML</a>, the second term equals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M199">View MathML</a>. It follows that for μ outside the zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M202">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M203">View MathML</a>

(3.4)

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M204">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M137">View MathML</a> let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M206">View MathML</a> be the squares determined by the vertices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a>. These squares do not intersect due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M209">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M210">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M211">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M212">View MathML</a>, it follows from the periodicity of tan that the number

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M213">View MathML</a>

is positive and independent of ε. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M214">View MathML</a> uniformly in the strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M215">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M216">View MathML</a>, there is an integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M217">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M218">View MathML</a>

By periodicity, there are numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M220">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M222">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M223">View MathML</a> and all k. Observing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M224">View MathML</a>, it follows that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M225">View MathML</a> such that for all μ on the squares <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M226">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M227">View MathML</a> the estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M228">View MathML</a> holds. Further, we may assume by Lemma 3.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M125">View MathML</a> is inside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M226">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M231">View MathML</a> and that no other zero of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> has this property. Hence, it follows by Rouché’s theorem that there is exactly one (simple) zero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M194">View MathML</a> of ϕ in each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M234">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M235">View MathML</a>. Replacing μ with only changes the sign of the second term in (3.3) and thus the sign of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M236">View MathML</a>. Hence, the same estimates apply to corresponding squares along the other three half-axes, and we therefore have that ϕ has zeros <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M237">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M238">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M239">View MathML</a> with the same asymptotic behavior as the zeros <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M241">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> as discussed in Lemma 3.1.

Next, we are going to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M236">View MathML</a> on the squares <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M245">View MathML</a>, whose vertices are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M246">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M248">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M249">View MathML</a>

(3.5)

Therefore, we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M250">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252">View MathML</a>, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M253">View MathML</a>

(3.6)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M256">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M257">View MathML</a>

uniformly in y as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M258">View MathML</a>. Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M259">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M260">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M263">View MathML</a>

(3.7)

It follows from (3.6) and (3.7) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M268">View MathML</a>

(3.8)

Furthermore, we are going to make use of the estimates

(3.9)

(3.10)

which hold for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M261">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M252">View MathML</a>. Therefore, it follows from (3.6), (3.8)-(3.10) and the corresponding estimates with μ replaced by that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M274">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M228">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M276">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M277">View MathML</a>. Again from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M236">View MathML</a> in (3.4) and Rouché’s theorem, we conclude that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> and ϕ have the same number of zeros in the square <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M281">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M282">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M284">View MathML</a> zeros inside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M286">View MathML</a> zeros inside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M287">View MathML</a>, it follows that ϕ has no large zeros other than the zeros <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M288">View MathML</a> found above for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M289">View MathML</a> sufficiently large, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M193">View MathML</a> account for all eigenvalues of the problem (2.1)-(2.2) since each of these eigenvalues gives rise to two zeros of ϕ, counted with multiplicity. By Proposition 2.3, all eigenvalues with non-zero real part occur in pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M292">View MathML</a>, which shows that we can index all such eigenvalues as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M190">View MathML</a>. Since there is an odd number of remaining indices, the number of pure imaginary eigenvalues must be odd.

Case 2: The function ϕ in this case is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M294">View MathML</a>

Then all the estimates are as in Case 4, and the result in Case 2 immediately follows from that in Case 4 if we observe that each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244">View MathML</a> for k large enough contains two fewer zeros of ϕ than in Case 4.

Case 1: A straightforward calculation gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M296">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M297">View MathML</a>

The result follows with reasonings and estimates as in the proof of Case 4, replacing μ by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M298">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M299">View MathML</a>, respectively.

Case 3: The function ϕ in this case is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M300">View MathML</a>

and a reasoning as in Case 1 completes the proof. □

4 Birkhoff regularity

We refer to [[15], Definition 7.3.1] for the definition of the Birkhoff regularity.

Proposition 4.1The boundary value problem (2.1), (2.5)-(2.8) is Birkhoff regular for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M301">View MathML</a>with respect to the eigenvalue parameterμgiven by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M302">View MathML</a>.

Proof The characteristic function of (2.1) as defined in [[15], (7.1.4)] is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M303">View MathML</a>, and its zeros are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M305">View MathML</a>. We can choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M306">View MathML</a>

according to [[15], Theorem 7.2.4.A]. The boundary condition (2.5)-(2.8) can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M307">View MathML</a>

where

and where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M309">View MathML</a> denotes the νth unit vector in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M310">View MathML</a>. Thus the boundary matrices defined in [[15], (7.3.1)] are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M311">View MathML</a>

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M312">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M313">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M314">View MathML</a>

for Case r and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M315">View MathML</a> for Cases 1 and 2, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M316">View MathML</a> for Cases 3 and 4. The Birkhoff matrices are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M317">View MathML</a>

(4.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M319">View MathML</a> are the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M320">View MathML</a> diagonal matrices with 2 consecutive ones and 2 consecutive zeros in the diagonal in a cyclic arrangement, see [[15], Definition 7.3.1 and Proposition 4.1.7]. It is easy to see that after a permutation of columns, the matrices (4.1) are block diagonal matrices consisting of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M321">View MathML</a> blocks taken from two consecutive columns (in the sense of cyclic arrangement) of the first two rows of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M322">View MathML</a> and the last two rows of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M323">View MathML</a>, respectively. Hence the determinants of the Birkhoff matrices (4.1) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M324">View MathML</a>

in Cases 1 and 2, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M325">View MathML</a>, whereas

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M326">View MathML</a>

in Cases 3 and 4. Thus, the problem (2.1), (2.5)-(2.8) is Birkhoff regular. □

5 Asymptotic expansions of eigenvalues

Let D, as a function of μ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M99">View MathML</a>, be the characteristic function of the problem (2.1), (2.5)-(2.8) with respect to the fundamental system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M319">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M330">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M331">View MathML</a>, where δ is the Kronecker delta. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332">View MathML</a> the corresponding characteristic function for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a>. Note that the characteristic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M335">View MathML</a> considered in Section 3 have the same zeros counted with multiplicity. Due to the Birkhoff regularity, g only influences lower order terms in D. Therefore, it can be inferred that outside the interior of the small squares <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M337">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M338">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M339">View MathML</a> around the zeros of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M341">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M342">View MathML</a> is sufficiently large. Since the fundamental system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M319">View MathML</a>, depends analytically on μ, also D and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M332">View MathML</a> are analytic functions. Hence, applying Rouché’s theorem both to the large squares <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M244">View MathML</a> and to the small squares which are sufficiently far away from the origin, it follows that the eigenvalues of the boundary value problem for general g have the same asymptotic distribution as for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M2">View MathML</a>. Whence Proposition 3.2 leads to

Proposition 5.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M348">View MathML</a>, there exists a positive integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M184">View MathML</a>such that the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M185">View MathML</a>, counted with multiplicity, of the problem (2.1), (2.5)-(2.8), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M186">View MathML</a>in Cases 1 and 2 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a>in Cases 3 and 4, can be enumerated in such a way that the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M353">View MathML</a>are pure imaginary for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M189">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M355">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M191">View MathML</a>. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M192">View MathML</a>, we can write<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M358">View MathML</a>, where the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M359">View MathML</a>have the following asymptotic representation as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195">View MathML</a>:

In particular, the number of pure imaginary eigenvalues is even in Cases 1 and 2 and odd in Cases 3 and 4.

In the remainder of the section, we are going to establish more precise asymptotic expansions of the eigenvalues. According to [[15], Theorem 8.2.1], (2.1) has an asymptotic fundamental system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M362">View MathML</a> of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M363">View MathML</a>

(5.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M364">View MathML</a>

(5.2)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M365">View MathML</a> means that we omit those terms of the Leibniz expansion which contain a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M366">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M367">View MathML</a>. Since the coefficient of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M368">View MathML</a> in (2.1) is zero, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M369">View MathML</a>, see [[15], (8.2.3)].

We will now determine the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M370">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M371">View MathML</a>. In this regard, observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M372">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M373">View MathML</a> in the notation of [[15], (8.1.2) and (8.1.3)], see [[15], Theorem 8.1.2]. From [[15], (8.2.45)], we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M374">View MathML</a>

(5.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M309">View MathML</a> is the νth unit vector in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M376">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M377">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M378">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M320">View MathML</a> matrices given by [[15], (8.2.28), (8.2.33) and (8.2.34)], that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M380">View MathML</a>,

(5.4)

(5.5)

(5.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M384">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M385">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M386">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M387">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M388">View MathML</a>. A lengthy but straightforward calculation gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M389">View MathML</a>

(5.7)

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M390">View MathML</a>

(5.8)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M391">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M392">View MathML</a> means that the estimate is uniform in x.

The characteristic function of (2.1), (2.5)-(2.8) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M393">View MathML</a>

where

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M395">View MathML</a>

(5.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M396">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M397">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M398">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M399">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M400">View MathML</a>, and each of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M401">View MathML</a> has asymptotic representations of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M402">View MathML</a>.

It follows from (5.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M403">View MathML</a>

(5.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M404">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M405">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M406">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M407">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M408">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M409">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M410">View MathML</a> and the terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M411">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M410">View MathML</a> can be absorbed by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M413">View MathML</a> as they are of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M414">View MathML</a> for any integer s. Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M415">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M416">View MathML</a>

(5.11)

where

(5.12)

(5.13)

A straightforward calculation gives

(5.14)

(5.15)

For the other two factors in (5.12) and (5.13), we have to consider the four different cases.

Therefore,

(5.16)

(5.17)

Thus, we have

(5.18)

(5.19)

Hence, we get

(5.20)

(5.21)

Thus, we have

(5.22)

(5.23)

We already know by Proposition 5.1 that the zeros <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M359">View MathML</a> of D satisfy the asymptotic representations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M434">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195">View MathML</a>. In order to improve on these asymptotic representations, write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M436">View MathML</a>

(5.24)

Because of the symmetry of the eigenvalues, we will only need to find the asymptotic expansions as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M195">View MathML</a>. We know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M438">View MathML</a> from Proposition 5.1, and it is our aim to find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M439">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M440">View MathML</a>. To this end, we will substitute (5.24) into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M441">View MathML</a> and we will then compare the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M443">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M444">View MathML</a>.

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M445">View MathML</a>

(5.25)

while

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M446">View MathML</a>

(5.26)

Using (5.11), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M441">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M448">View MathML</a>

(5.27)

where γ is the highest μ-power in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M413">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M450">View MathML</a>. Substituting (5.25) and (5.26) into (5.27) and comparing the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M442">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M443">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M444">View MathML</a>, we get

Theorem 5.2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M348">View MathML</a>, there exists a positive integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M184">View MathML</a>such that the eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M456">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a>, counted with multiplicity, of the problem (2.1), (2.5)-(2.8), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M458">View MathML</a>in Cases 1 and 2 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M162">View MathML</a>in Cases 3 and 4, can be enumerated in such a way that the eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M456">View MathML</a>are pure imaginary for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M189">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M462">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M191">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M464">View MathML</a>and the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M465">View MathML</a>have the asymptotic representations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M466">View MathML</a>

and the numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M438">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M439">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M440">View MathML</a>are as follows:

In particular, the number of pure imaginary eigenvalues is even in Cases 1 and 2 and odd in Cases 3 and 4.

Remark 5.3 In [14] we have considered the differential equation (2.1) with the same boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M471">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M472">View MathML</a> at a as in this paper but with λ-independent boundary conditions at 0, that is, the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M473">View MathML</a> also occur in [14]. Whereas in [14] the number of pure imaginary eigenvalues is odd in each case, this number is even in Cases 1 and 2 of this paper. We observe that in Cases 1 and 2, the λ-dependent part is the ‘dominating’ part of the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M474">View MathML</a>, in the sense that it has the highest μ-power arising as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M475">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M476">View MathML</a>, whereas in Cases 3 and 4 the λ-independent part is dominating. It may be interesting to investigate if, in general, the parity of the number of pure imaginary eigenvalues can be determined by the number of dominating λ-dependent parts in the boundary conditions.

We can observe that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M118">View MathML</a> in the Cases 3 and 4 are respectively the same as in [14] since the corresponding dominating terms in the boundary conditions coincide. However, the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M439">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M440">View MathML</a> differ from those of [14] in each case, which is due to the λ-term in the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/106/mathml/M474">View MathML</a>.

Competing interests

The authors do not have any competing interests.

Authors’ contributions

The subject of this paper is part of the PhD thesis of BZ. The subject has been suggested and supervised by MM, and the initial version of the paper has been written by BZ. The submitted version has been verified and discussed by MM and BZ.

Acknowledgements

This research was partially supported by a grant from the NRF of South Africa, Grant number 69659. Various of the above calculations have been verified with Sage.

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