Eigenvalues of Second-Order Left-Definite Linear Difference Operator with Spectral Parameters in Boundary Conditions

Abstract

In this paper, the authors consider the spectra of second-order left-definite difference operator with linear spectral parameters in two boundary conditions. First, they obtain the exact number of this kind of eigenvalue problem, and prove these eigenvalues are all real and simple. In details, they get that the number of the positive (negative) eigenvalues is related to not only the number of positive (negative) elements in the weight function, but also the parameters in the boundary conditions. Second, they obtain the interlacing properties of these eigenvalues and the sign-changing properties of the corresponding eigenfunctions according to the relations of the parameters in the boundary conditions.

Keywords

Left-definite difference operator / Boundary conditions with spectral parameters / Interlacing properties / Oscillation properties