Criteria for Discrete Spectrum of 1D Operators

作     者：Mu-Fa Chen 

作者机构：School of Mathematical SciencesBeijing Normal UniversityLaboratory of Mathematics and Complex Systems(Beijing Normal University)Ministry of EducationBeijing 100875People’s Republic of China 

出 版 物：《Communications in Mathematics and Statistics》 (数学与统计通讯（英文）)

年 卷 期：2014年第2卷第3期

页      面：279-309页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：The author thanks S.Kotani for introducingandto him and R.O˘ınarov for sending him the original version of.Thanks are also given to H.J.Zhang and Z.W.Liao for their corrections of an earlier version of the paper.Research supported in part by the National Natural Science Foundation of China(No.11131003) the“985”project from the Ministry of Education in China,and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions 

主　　题：Discrete spectrum Essential spectrum Tridiagonal matrix(birth–death process) Second-order differential operator(diffusion) Killing 

摘      要：For discrete spectrum of 1D second-order differential/difference operators(with or without potential(killing),with the maximal/minimal domain),a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the ***,these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier,simply replacing the‘finite supremum’by‘vanishing at infinity’.Except a dual technique,the main tool used here is a transform in terms of the harmonic function,to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous *** of them are illustrated by *** main body of the paper is devoted to the hard part of the story,the easier part but powerful one is delayed to the end of the paper.