Characterization Of Eigenvalues For Singular Differential Operators

The theory of ordinary differential operators can supply the theoretical basis for differential equations, modern physics and other technique fields. It investigates a great deal of important problems, such as spectral analysis, self-adjoint extension, deficiency index theory, completeness of eigenfunctions, inverse questions and so on.The researches of ordinary differential operator originated from the solid heat transfer problem and variety of classical mathematics physics definite solutions in the early19th century. The self-adjointness of differential operator is an important part of differential operator theory and it draws much interest of mathematical researchers.The research of Hamiltonian systems, it is the core of the differential operator. It originates from the fields of mathematical science, life science and so on, especially from many mathematical models of quantum methanics and biological engineering.On the one hand, this paper obtains the method of eigenvalues of the points which satisfies some conditions in the spectral gap of the minimal operator generated by singu-lar differential operator, and puts on the process of self-adjoint of operator by spectral gap. On the other hand, a necessary and sufficient condition of the Hamiltonian system can be obtained.The thesis is divided into three sections according to its content.Chapter1Preference, we introduce the main contents of this paper.Chapter2In this part, we consider the spectral properties for n order differential operator which defined on (α,β) Given a spectral gap (a, b) of the minimal operator To which is generated by τ with deficiency index being equal to r, arbitrary m points βi(i=1,2,…,m,m≤r) in (a,b), and a positive integer function p which is satisfied∑im=1p(βi)≤r,T0has a self-adjoint extension T such that each βi(i=1,2,…,m) is an eigenvalue of T with multiplicity at least p(βi). Chapter3In this part,we consider the limit circle case of the liner Hamiltonian system Jy’(t)-H(t)y(t)-λW(t)y(t),t∈I-[a,b),(α is a regular point and b is a singular point)and we obtain a necessary and sufficient condition of the Hamiltonian system.