On Discreteness Of Spectrum Of Differential Operators And Discrete Spectrum Analysis

In this paper, we investigate the discreteness of spectrum of differen-tial operators, the changing tendency of eigenvalues as boundary condition parameter of self-adjoint singular Sturm-Liouville problems varies, spec-tral properties of Sturm-Liouville problem with eigen-parameter dependent boundary conditions and its regular approximation.The spectrum of self-adjoint differential operators generated by2n or-der symmetric differential expression with real coefficients are real and in general have not only discrete spectrum, but also essential spectrum. Since any self-adjoint extension of the minimal operator has the same essential spectrum, thus the discreteness of spectrum is only related to the coeffi-cients of differential equations and is not related to self-adjoint boundary conditions. Since in1953Molchanov published his celebrated criterion on the discreteness of the second order Sturm-Liouville operator, the discrete-ness of spectrum has been paid more attention and some useful results are obtained. However the problem of the discreteness is not completely solved. In this paper we will give some new conditions of the discreteness of spectrum of differential operators from some different aspects by means of the compactness of Sobolev spaces, the decomposition principle or os-cillation theory of differential equations. Using the results obtained, the discreteness of spectrum of some differential operators is easily judged.For Sturm-Liouville problem, either regular case or limit circle non- oscillation case, eigenvalues have corresponding changes as the boundary condition parameter varies. For Sturm-Liouville problem with one limit point endpoint, the essential spectrum are independent on the boundary condition parameters but the discrete spectrum are dependent on them. Generally the boundary condition parameters have an impact not only on the value of eigenvalues but also on the existence of eigenvalues. Using the oscillation theory of the eigenfunctions, spectral theorem as well as regular approximation of eigenvalues of singular differential operators, we discuss the existence of eigenvalues below the essential spectrum, give the con-tinuous and differentiable dependence of eigenvalues on boundary condi-tions parameter, and get some eigenvalues inequalities among the different boundary conditions. These results play an important role in the further study of spectral property and provide significant theoretical foundation for numerical computation of eigenvalues.Furthermore, Sturm-Liouville problems with eigenparameter depen-dent boundary conditions are considered. The essential spectrum coincide with those of Sturm-Liouville problem with eigenparameter independent boundary conditions. And the inherited boundary conditions and induced restriction operators are constructed and the regular approximation of op-erators and eigenvalues is derived. For the case with two singular endpoints and eigenparameter dependent boundary conditions, we construct the new Hilbert space and the new operator, prove that the operator constructed is self-adjoint in the Hilbert space and then obtain some spectral properties in terms of the new space, the new operator together with the theory of self-adjoint operators in Hilbert spaces.This paper contains seven parts. The first is an introduction of the background of the problems we investigate and main results of this paper. The second is Sobolev embedding and the discreteness of spectrum of differ-ential operators. The third is decomposition principle and the discreteness of spectrum of differential operators. The fourth is oscillation of differen-tial equations and the discreteness of spectrum. The fifth is eigenvalues of singular Sturm-Liouville problem. The sixth is singular Sturm-Liouville problem with eigenparameter dependent boundary conditions in an regu-lar endpoint and a singular endpoint and its approximation. And the last one is singular Sturm-Liouville problem with two singular endpoints and eigenparameter dependent boundary conditions.