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Coupling constant limits of Schrödinger operators with critical potentials

Xiaoyao Jia1* and Yan Zhao2

Author Affiliations

1 Mathematics and Statistics School, Henan University of Science and Technology, No. 263, Luo-Long District, Kai-Yuan Road, Luoyang City, Henan Province, 471023, China

2 College of Civil Engineering and Architecture, Zhejiang University, B505 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang Province, 310058, China

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Boundary Value Problems 2013, 2013:62  doi:10.1186/1687-2770-2013-62

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/62


Received:8 December 2012
Accepted:4 March 2013
Published:27 March 2013

© 2013 Jia and Zhao; 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 family of Schrödinger operators, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M1">View MathML</a>, is studied in this paper. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M3">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M4">View MathML</a> is large enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M5">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M6">View MathML</a>. We show that each discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> tends to 0 when λ tends to some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a>. We get asymptotic behavior of the smallest discrete eigenvalue when λ tends to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a>.

Keywords:
Schrödinger operator; critical potential; asymptotic expansion

1 Introduction

In this paper, we consider a family of Schrödinger operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> which are the perturbation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11">View MathML</a> in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M12">View MathML</a>

on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M14">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M15">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M16">View MathML</a> are the polar coordinates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M17">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M18">View MathML</a> is a real continuous function. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M19">View MathML</a> is a non-zero continuous function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M20">View MathML</a>

(1)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M21">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M22">View MathML</a> denote the Laplace operator on the sphere <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M23">View MathML</a>. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M24">View MathML</a>

(2)

If (2) holds, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M25">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a> (see [1]).

Under the assumption on V, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> has discrete eigenvalues when λ is large enough, and each discrete eigenvalue tends to zero when λ tends to some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a> (see Section 2). We study the asymptotic behaviors of the discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> in this paper. The asymptotic behaviors for Schrödinger operators with fast decaying potentials were studied by Klaus and Simon [2]. In [2], they studied the convergence rate of discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M30">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M31">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a> is the value at which some discrete eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M33">View MathML</a> tends to zero. The main method they used in their paper is the Birman-Schwinger technique.

In order to use the Birman-Schwinger technique to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>, we need to get the asymptotic expansion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M35">View MathML</a> for α near zero, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M36">View MathML</a>, which was studied by Wang [1]. In this paper, we first show that there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a> such that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> has discrete eigenvalues. Then, we define the Birman-Schwinger kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> and find that there is one-to-one correspondence between the discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a> and the discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>. Hence, the asymptotic expansion of the discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> can be got through the asymptotic expansion of the discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a>. In our main results, we need to use that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a> is a bounded operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47">View MathML</a>. To get that, we add a strong condition on V (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M49">View MathML</a> in (1)). We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a> is a family of compact operators converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51">View MathML</a> and obtain the asymptotic expansions of the discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a> by functional calculus. After that, the convergence rate of the smallest discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> is obtained.

Here is the plan of our work. In Section 2, we recall some results of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11">View MathML</a> and define the Birman-Schwinger kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>. The relationship between the eigenvalues of these two kinds of operators is studied. In Section 3, we first study the asymptotic behavior of the discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a>. Then the convergence rate of the smallest discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> is obtained. We get the leading term and the estimate of the remainder term of the smallest discrete eigenvalue.

Let us introduce some notations first.

Notation The scalar product on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M61">View MathML</a> and that on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M62">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M63">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M66">View MathML</a>, denotes the weighted Sobolev space of order r with volume element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M67">View MathML</a>. The duality between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M69">View MathML</a> is identified with the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47">View MathML</a> product. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M71">View MathML</a>. Notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M72">View MathML</a> stands for the space of continuous linear operators from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M73">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M74">View MathML</a>. The complex plane ℂ is slit along positive real axis so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M76">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M77">View MathML</a> are holomorphic there.

2 Some results for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11">View MathML</a>

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M16">View MathML</a> are the polar coordinates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M17">View MathML</a>. Then the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M81">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M82">View MathML</a>

(3)

in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a> (see [1]).

Now, we recall some results on the resolvent and the Schrödinger group for the unperturbed operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M85">View MathML</a>

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M86">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M87">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M88">View MathML</a> denote the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M89">View MathML</a> as the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M90">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M93">View MathML</a> denote an orthogonal basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M94">View MathML</a> consisting of eigenfunctions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M90">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M96">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M97">View MathML</a> denote the orthogonal projection in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M94">View MathML</a> onto the subspace spanned by the eigenfunctions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M90">View MathML</a> associated with the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M100">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M101">View MathML</a> denote the orthogonal projection in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M94">View MathML</a> onto the eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M103">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M104">View MathML</a>

Denote for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M87">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M106">View MathML</a>

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M107">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M108">View MathML</a> is the largest integer which is not larger than ν. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M109">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M110">View MathML</a> be the largest integer strictly less than ν. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M111">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M112">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M113">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M114">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M116">View MathML</a>, otherwise. One has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M117">View MathML</a>.

The following is the asymptotic expansion for the resolvent <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M118">View MathML</a>.

Theorem 2.1 (Theorem 2.2 [1])

The following asymptotic expansion holds forznear zero with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M119">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M120">View MathML</a>

in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M122">View MathML</a>. Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M123">View MathML</a>

Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M124">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M125">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M126">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M127">View MathML</a>a polynomial inρof degree j:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M128">View MathML</a>

First, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> has discrete eigenvalues when λ is large enough. In fact, we need only to show that there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M130">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M131">View MathML</a>.

From the assumption on V, we know that there exists a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M132">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M133">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M134">View MathML</a> small enough such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M136">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M139">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M140">View MathML</a>

when λ is large enough, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M141">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> has discrete eigenvalues when λ is large enough.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> has a continuous spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M144">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M145">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M146">View MathML</a> exists and equals zero (see [3]). We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M147">View MathML</a>. Hence, from the continuity of a discrete spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>, we know that there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M8">View MathML</a> such that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> has eigenvalues less than zero, and when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M153">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> has an eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M155">View MathML</a> at the bottom of its spectrum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M156">View MathML</a>. In Section 3 (Proposition 3.1), we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> is simple and the corresponding eigenfunction can be chosen to be positive everywhere. (There are many results about the simplicity of the smallest eigenvalue of the Schrödinger operator without singularity, but there is no result which can be used directly, because the potential we use in this paper has singularity at zero. Theorem XIII.48 [4] can treat the Schrödinger operator with the potential which has singularity at zero, but the positivity of potential is needed. Hence, we give this result.) From the discussion above and the continuity of a discrete spectrum, one has that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> tends to zero at some λ. The asymptotic behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> is studied in this paper.

To study the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>, we first define a family of Birman-Schwinger kernel operators. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M161">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M162">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M163">View MathML</a>

Then we have the following result.

Proposition 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M36">View MathML</a>. Then

(a) Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M165">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M166">View MathML</a>is injective fromAtoB, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M167">View MathML</a>is injective fromBtoA.

(b) The multiplicity ofαas the eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>is exactly the multiplicity of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169">View MathML</a>as the eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a>.

Proof (a) First, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M166">View MathML</a> is injective from A to B. Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M172">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M173">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M174">View MathML</a>. And if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M175">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M176">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M166">View MathML</a> is injective from A to B.

Next, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M167">View MathML</a> is injective from B to A. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M179">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M180">View MathML</a>

And if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M181">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M182">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M167">View MathML</a> is injective from B to A.

(b) From (a), one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M184">View MathML</a>. This means that the multiplicity of α as the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> is exactly the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169">View MathML</a> as the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a>. □

From the last proposition, we know that there exists one-to-one correspondence between the discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> and the discrete eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a>. Hence, we can study the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a> first.

3 Asymptotic expansion of the eigenvalues

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M11">View MathML</a> and V are defined as above, we show that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M192">View MathML</a> has the eigenvalue less than zero, then the smallest eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M192">View MathML</a> is simple. We use Theorems XIII.44, XIII.45 [4] to prove it.

Proposition 3.1Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M192">View MathML</a>has an eigenvalue at the bottom of its spectrum. Then this eigenvalue is simple and the corresponding eigenfunction can be chosen to be a positive function.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M195">View MathML</a> be a smooth nonincreasing function such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M196">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M198">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M199">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M200">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M204">View MathML</a>. From the proof of Theorem XIII.47 [4], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M205">View MathML</a> is positivity preserving and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M206">View MathML</a> acts irreducibly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47">View MathML</a>. Hence, by Theorem XIII.45 [4], if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M208">View MathML</a> converges to H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M209">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M210">View MathML</a> in the strong resolvent sense, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M211">View MathML</a> is positivity preserving and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M212">View MathML</a> acts irreducibly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47">View MathML</a>. By Theorems XIII.43 and XIII.44 [4], we can get the result. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M214">View MathML</a> is the core for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M215">View MathML</a> and P, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M217">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M47">View MathML</a>, then we have the necessary strong resolvent convergence by Theorem VIII.25(a) [5]. This ends the proof. □

Proposition 3.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M220">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M221">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M223">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M222">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M225">View MathML</a>. For any test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M226">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M227">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M228">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M229">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M230">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M231">View MathML</a>. It follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M232">View MathML</a> because ϕ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M233">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M221">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M220">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M221">View MathML</a>. □

Proposition 3.3Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a>is a compact operator for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M239">View MathML</a>. And<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M40">View MathML</a>converges to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51">View MathML</a>in operator norm sense.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M243">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M244">View MathML</a> is a bounded operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M246">View MathML</a>, and V is a compact operator from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M246">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M249">View MathML</a> is a compact operator on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a>. Using a similar method to that in Proposition 2.2, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M249">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M252">View MathML</a> have the same non-zero eigenvalues, and for the same eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M253">View MathML</a>, the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M253">View MathML</a> as the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M255">View MathML</a> and the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M253">View MathML</a> as the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M252">View MathML</a> are the same. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M252">View MathML</a> is a compact operator. Because

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M259">View MathML</a>

and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M260">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M261">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M263">View MathML</a> in operator norm sense as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M264">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51">View MathML</a> is a compact operator. □

Lemma 3.4Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M266">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M267">View MathML</a>are two bounded self-adjoint operators on a Hilbert spaceH. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M268">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M269">View MathML</a>.

Proof By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M270">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M271">View MathML</a>

This ends the proof. □

Lemma 3.5Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a>is a family of compact self-adjoint operators on a separable Hilbert spaceH, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M273">View MathML</a>forαnear zero. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M274">View MathML</a>

Then:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275">View MathML</a>is an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275">View MathML</a>converges when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M278">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M279">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M280">View MathML</a>is an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>.

(b) Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M282">View MathML</a>is an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>of the multiplicity ofm. Then there aremeigenvalues (counting multiplicity), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M284">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M285">View MathML</a>), of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a>near<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287">View MathML</a>. Moreover, we can choose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M288">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M289">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M290">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M291">View MathML</a>is the eigenvector of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a>corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M293">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M294">View MathML</a>), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M291">View MathML</a>converges as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M278">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M291">View MathML</a>converges to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M298">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M298">View MathML</a>is the eigenvector of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287">View MathML</a>.

Proof (a) By the min-max principle, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275">View MathML</a> is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a>. By Lemma 3.4, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M304">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M275">View MathML</a> converges to the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>.

(b) Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a> is a compact operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M282">View MathML</a> is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287">View MathML</a> is a discrete spectrum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>. Then there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M312">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a> has only one eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M315">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M316">View MathML</a>). For α small enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a> has exactly m eigenvalues (counting multiplicity) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M315">View MathML</a> because the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a> converge to the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a> by part (a) of lemma. Suppose the m eigenvalues, near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287">View MathML</a>, of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M272">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M323">View MathML</a>, and the corresponding eigenvectors are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M324">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M325">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M326">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M327">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M328">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M329">View MathML</a>. For α near zero, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M330">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M331">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M298">View MathML</a> be an element in A such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M333">View MathML</a> acquires the minimum value. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M334">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M335">View MathML</a>

In the last equality, we use the fact

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M336">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M337">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M338">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M339">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M340">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M341">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M342">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M343">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M344">View MathML</a>. This ends the proof. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M345">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M346">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M347">View MathML</a> or 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M348">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M349">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M350">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M351">View MathML</a> are compact operators, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M352">View MathML</a> for β near zero. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M353">View MathML</a>

Then, by the min-max principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a> is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>. Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M356">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a> is a discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a> is a compact operator. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M356">View MathML</a> is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a> of multiplicity m, without loss, we can suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M362">View MathML</a>. Then there exist exactly m eigenvalues (counting multiplicity), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M363">View MathML</a>, of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364">View MathML</a> near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a>. By Lemma 3.5, we know that there exists a family of normalized eigenvectors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M366">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M368">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M369">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M370">View MathML</a>), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M372">View MathML</a>) converge as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M373">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M371">View MathML</a> converge to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M375">View MathML</a> for all j such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M376">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M377">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M378">View MathML</a> can be extended to a standard orthogonal basis. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M379">View MathML</a>

Then we have the following.

Lemma 3.6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a>are given as before. Then the eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M383">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M372">View MathML</a>) has the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M385">View MathML</a>

Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M386">View MathML</a>

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M356">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a> is the discrete eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M281">View MathML</a>. Suppose that the multiplicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a> is m, and suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M391">View MathML</a> as before. Hence, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M392">View MathML</a> small enough such that there is only one eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M394">View MathML</a>. We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M383">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M372">View MathML</a>) converge to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M354">View MathML</a>. It follows that if δ is small enough, there are exactly m eigenvalues (counting multiplicity) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M364">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M399">View MathML</a> for β small. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M400">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M401">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M402">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M403">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M404">View MathML</a>

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M405">View MathML</a>

In the last step, we use that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M406">View MathML</a>

Similarly, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M407">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M408">View MathML</a>

 □

First, we study the asymptotic expansion of the smallest eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>. By Proposition 3.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> is a simple eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>, and the corresponding eigenfunction can be chosen to be positive. We suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M413">View MathML</a> is a positive eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M415">View MathML</a>. Without loss of generality, we can suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M416">View MathML</a>. Then we can get the following result.

Lemma 3.7Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219">View MathML</a>. Set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M418">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M419">View MathML</a>is defined as above. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420">View MathML</a>converges in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M422">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M423">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420">View MathML</a>converges toϕ, thenϕis the eigenfunction of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M426">View MathML</a>.

Proof By the assumption of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M428">View MathML</a>. One can check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M13">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M431">View MathML</a> by Lemma 3.5. And also, by Lemma 3.5, we know that ϕ is the normalized eigenfunction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M51">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M287">View MathML</a>. ϕ is a positive function since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420">View MathML</a> is a positive function. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M435">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M436">View MathML</a> and u is a positive function because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M437">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M438">View MathML</a>

In the last equality, we use the fact that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M439">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M440">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M441">View MathML</a>. This ends the proof. □

Theorem 3.8Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M219">View MathML</a>. ϕis defined in Lemma 3.7. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M443">View MathML</a>, one of three exclusive situations holds:

(a) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M444">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M445">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M446">View MathML</a>.

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M447">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M448">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M449">View MathML</a>.

(c) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M450">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M451">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M452">View MathML</a>.

Proof

(a) By Theorem 2.1, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M453">View MathML</a>

Then if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M454">View MathML</a>, we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M455">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M456">View MathML</a>. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> is the simple eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169">View MathML</a> is the simple eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M19">View MathML</a>, one has that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M7">View MathML</a> is monotonous with respect to λ and so is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460">View MathML</a> and the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460">View MathML</a> are monotonous with respect to λ. Therefore, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M169">View MathML</a> is the biggest eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460">View MathML</a>. If not, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M468">View MathML</a> is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460">View MathML</a>, then by the continuity and monotony of the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460">View MathML</a> with respect to λ, we know that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M471">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M472">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M473">View MathML</a> is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M474">View MathML</a>. This is contradictory to that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> is the smallest eigenvalue. By Lemma 3.7, we know the normalized eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M476">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M460">View MathML</a> converges to ϕ. It follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M478">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M479">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M480">View MathML</a>

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M481">View MathML</a> is the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M482">View MathML</a> corresponding to the eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M420">View MathML</a>. By Lemma 3.6, we should compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M484">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M485">View MathML</a>. From the definition of ϕ, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M486">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M487">View MathML</a>

In the last equality, we use the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M488">View MathML</a>, which can be obtained by Proposition 3.2. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M489">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M130">View MathML</a> by Theorem 3.1 [1]. So, ψ is the ground state of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M491">View MathML</a>. We also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M492">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M493">View MathML</a>

It follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M494">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M495">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M496">View MathML</a>. By the Proposition 2.2, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M497">View MathML</a>. It follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M498">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M499">View MathML</a>, we can get the leading term of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M501">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M502">View MathML</a>.

(b) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M503">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M504">View MathML</a>

By Lemma 3.7, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M505">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M506">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M507">View MathML</a>. As in (a), using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M508">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M509">View MathML</a>

one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M510">View MathML</a>. To get the leading term of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M157">View MathML</a>, we can suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M512">View MathML</a>. Then, by comparing the leading term, we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M513">View MathML</a>. It follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M514">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M515">View MathML</a>.

(c) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M516">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M517">View MathML</a>

By Lemma 3.7, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M518">View MathML</a>. Using the same argument as before, we can conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M519">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M520">View MathML</a>. As above, we can get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M521">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/62/mathml/M522">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This research is supported by the Natural Science Foundation of China (11101127,11271110) and the Natural Science Foundation of Educational Department of Henan Province (2011B110014).

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