Invariance Of Defect Indices And Spectra Of Hermitian Subspaces Under Perturbations And Their Applications

Perturbation theory was established by L. Rayleigh and E. Schrodinger. L. Rayleigh [1] gave a formula for computing the natural frequencies and mode of a vibrating system deviating slightly from a simpler system which admits a complete determination of the frequencies and modes. Mathematically speak-ing, the method is to get approximate solutions of the perturbed eigenvalue problem by the solutions of perturbed eigenvalue problem that is simpler and can be easily solved.E. Schrodinger [2] developed a similar method, with more generality and systematization, for the eigenvalue problems that, appear in quantum mechanics.Later, F. Rellich, J. D. Newburgh, F. Wolf. T. Kato and B. Simon developed the perturbation theory of linear operators later [3-7].The perturbation theory of differential operators have been investigated deeply, and some good results have been obtained.In1987, J. Weidmann studied stability of limit cases of singular second-order differential operators under relatively bounded perturbation with bound less than1[8]. In1997, T. G. Anderson and D. B. Hinton gave necessary and sufficient conditions for perturbations of a second-order ordinary differential operator to be either rela-tively bounded or relatively compact [9]. In1998, the necessary and sufficient. conditions were obtained for a perturbed operator to be relatively bounded or relatively compact for a classes of nth order differential operators by T. G. Anderson [10].The perturbation theory of difference operators have been also investi-gated. In2003, J. Chen and Y. Shi studied invariance of the limit cases of singular second-order difference operators when the weight function is equal to1and the potential function is perturbed by a bounded function [11]. In2012, Z. Zheng obtained that the minimal and maximal deficiency indices are invariant under constantly bounded perturbation for discrete Hamiltonian systems [12]. However, for a symmetric linear difference equation, the corre-sponding minimal operator is non-densely defined and multi-valued, and the maximal operator is in general a multi-valued operator. So the perturbation theory of symmetric operators is not available in the study of difference equa-tions. In order to solve this problem, Arens, Coddington, Lesch, Malamud and other scholars have successfully extended the concepts and some results of densely defined Hermitian operators to Hermitian subspaces [13-24]. In2012, Y. Shi [25] extended the classical GKN theory to Hermitian subspace, and based on this, she with H. Sun gave out a complete characterization of self-adjoint extensions for second-order symmetric linear difference equation [26]. Recently, Shi, Shao, Ren and Liu studied some spectral properties and resolvents of self-adjoint subspaces [27-28], which has laid a foundation for the study of perturbation problems about difference equationsThis thesis is devoted to invariance of defect indices and spectra of Hermi-tian subspaces under perturbations and their application to singular second-order difference operators and discrete linear Hamiltonian systems.This thesis is divided into three chapters.In Chapter1, Section1is introduction. In Section2, some basic concepts and fundamental theory of subspaces are introduced. We firstly give some ba-sic concepts, including relatively bounded subspaces and essential self-adjoint subspaces. Later, we give a necessary and sufficient condition for an Hermitian subspace to be essential self-adjoint. In Section3, we obtain some sufficient conditions for invariance of defect indices of Hermitian subspaces and spectra of non-self-adjoint closed Hermitian subspaces under perturbations, and some necessary and sufficient conditions for a self-adjoint subspaces to have a pure discrete spectrum. In Section4, we also give out a minimax theorem of semi-bounded self-adjoint subspaces, and a sufficient condition for a semibounded self-adjoint subspace to have a pure discrete spectrum under perturbation.In Chapter2, Section1is introduction. In Section2, some basic concepts and fundamental theory of second-order symmetric linear difference equations are introduced. In Section3, we pay attention to proving that in the limit circle case every self-adjoint subspace extension of minimal subspace of a singular second-order difference equation has a pure discrete spectrum in terms of the corresponding Green function. In Section4. we study how the coefficients of a second-order difference equation changes such that its defect indices are invariant. We firstly consider the case that the leading coefficient and potential function are perturbed but the weight function is always equal to1. Then we study the case that the weight function is perturbed while the leading coefficient and potential function don’t be changed. Finally, we discuss the general case that the three terms are all perturbed at the same time.In Chapter3, Section1is introduction. In Section2, some basic concepts and fundamental theory of discrete linear Hamiltonian systems are introduced. In Section3, we shall first construct the Green function of singular discrete lin-ear Hamiltonian systems in the limit circle case, then use it to prove that every self-adjoint subspace extension of its minimal subspace have a pure discrete spectrum. In Section4, we shall focus on studying invariance of defect indices of a singular discrete linear Hamiltonian system in the case that the potential function is perturbed, while the weight function does not be changed. In par-ticular, we consider invariance of the limit point case and limit circle case of a singular discrete linear Hamiltonian system under perturbation of the weight function and potential function.