Spectra of a discrete Sturm-Liouville problem with eigenparameter-dependent boundary conditions in Pontryagin space

Abstract

In this paper, we consider the spectra of a discrete Sturm-Liouville problem with two eigenparameter-dependent boundary conditions. Different from the previous results, the operator corresponding to the problem here is not self-adjoint in the corresponding Hilbert space, but J-self-adjoint in a new Pontryagin space under a new J-metric. No matter what, the existence, simplicity and interlacing properties of real eigenvalues, nonreal eigenvalues and the oscillation properties of eigen functions are obtained in this paper. Meanwhile, we also obtain the existence of a pair of nonreal eigenvalues and an unconditional basis of the Pontryagin space under the new J-metric.

Key words: Discrete Sturm-Liouville problems, eigenparameter-dependent boundary conditions, interlacing, oscillation properties, Pontryagin space.