The Spectral Analysis Of Several Differential Operators

In this paper, we investigate the spectrum of discontinuous singular differential op-erators and the discreteness of spectrum for several differential operators with special coefficients.Firstly, the singular Sturm-Liouville problem with eigenparameter dependent bound-ary condition at regular endpoint and with transmission conditions at one inner point of a finite interval is obtained with operator method and function theory. A new direct sum Hilbert space be defined by the new product associated with transmission conditions. In this space the considered problem is converted the corresponding singular operator and the singular operator is self-adjoint. Its eigenvalues coincide with eigenvalues of Sturm-Liouville problem. Then we prove that the eigenvalues are real, at most countable, bounded below and they are the zeros of its discriminant function. And we give asymp-totic formulas for its eigenvalues. Furthermore the discontinuous singular Sturm-Liouville problems with eigenparameter dependent two boundary conditions are considered. The problems become the corresponding operator problems with constructing new Hilbert space and the self-adjointness of the problems are got. And the eigenvalues coincide with the zeros of its discriminant function and asymptotic formulas for the eigenvalues are investigated.Secondly, we consider the spectrum of differential operators for symmetric differential expressions with real power-exponential product coefficients and real Euler-exponential product coefficients. If coefficients of differential expresses satisfy certain conditions, the spectrum of such operators is discrete with methods of direct sum decomposition for operators and quadratic comparison. And the discreteness of spectrum for differential op-erator which is generated by symmetric differential with general coefficients is discussed. We further find that the discreteness of the spectrum of such operators not only deter-mined by the last term and first term coefficient whose limits are infinity in some certain way, but also the middle term coefficient whose limits are infinity with certain way also can decide the discreteness of the spectrum.Finally, the spectrum of a class of even order symmetric differential operator with complex exponential coefficient be discussed. The spectrum of the considered differen-tial operator is discrete when the real part and imaginary parts of its coefficients are non-negative. Furthermore the spectrum is discrete if the real and imaginary parts of its coefficients satisfy the general appropriate conditions. And then the discreteness of spectrum for differential operators are investigated which be generated by J-symmetric differential expressions with complex power-exponential product coefficient and complex Euler-exponential product coefficient respectively. The essential spectrum of any J-self-adjoint extensions for the minimal operator is empty when the real and imaginary parts of those coefficients satisfy some conditions, i.e. the spectrum of any J-self-adjoint extensions is discrete.This thesis is divided into seven chapters. The backgrounds and the main results of the discussed questions are described in chapter1. In Chapter2, we give the related fundamental concepts and properties involved in this thesis; The singular Strum-Liouville problem with eigenparameter dependent boundary conditions at regular endpoint and transmission conditions is considered in Chapter3. Chapter4investigates a class of discontinuous singular Strum-Liouville problem with eigenparameter dependent bound-ary conditions. The discontinuous singular Strum-Liouville problem with eigenparameter dependent boundary conditions and with transmission conditions at finite inner points of considered interval be investigated in Chapter5. Chapter6study the discreteness of the spectrum on several even order differential operators with special coefficients and the discreteness of the spectrum for several J-symmetric differential operators with special coefficients are given in Chapter7.