On The Research Of Spectrum Of Differential Operator

In this paper, we study an important problem in the field of differentialoperators: spectral analysis. First we consider the asymptotic expansion ofeigenvalues for a regular Sturm-Liouville Problem on [0, π]. By combiningFrechet derivative, we give a sophisticated analysis for the eigenvalues, whichreveals the explicit effects of Constant of boundary condition and equationcoefficient.In the second part of this paper, We study symmetric differentialexpressionDiscusses the discreteness of the spectrum of differential operatorsgenerated by above differential expression and the coefficients ak (x)is thereal-valued functions, we give the sufficient and necessary conditions fordiscreteness of the spectrum, and give the sufficient and nessessaryconditions for discreteness of the spectrum of Euler differential operators.This paper contains three parts. The first part is an introduction of thebackground and main results of this paper. The second part discussesasymptotic expansion of eigenvalues for a regular Sturm-Liouville Problem. The third part discusses the discreteness of the spectrum of differentialoperators.