Spectral Problems Of Discontinuous Left Definite Differential Operators

In this paper, spectral problems of discontinuous left definite differential operators will be considered.The paper is mainly divided into four chapters.The first section is the introduction of the whole paper. We firstly mention the results of former scholars about this problem. Then we introduce the problems of this paper and give conclusions.Next section of the paper mainly investigates the following left definite spectral problems of discontinuous differential equationWe can obtain the spectral properties of the self-adjoint operators by the analysis to the discontinuous conditions and the boundary value conditions and the constructing of the proper Hilbert space .The third section mainly investigates the following discrete Sturm-Liouville systemwith the boundary value conditionwhere K is a symmetric and positive definite 2×2 matrix,ω_n can change sign and the coefficient p_n-1 is allowed to be 0. Under these conditions, we will give the existence and distribution of the eigenvalues of this problem.At last, we summarize the results of the whole paper.