Spectral analysis of a difference operator with a growing potential

Abstract

In this paper, we study the spectral properties of a second order difference operator with a growing potential. The operator acts in the complex Hilbert space ℓ2(Z) of square summable complex sequences indexed by the integers. This operator is a discrete analogue of a second order differential operator with a growing complex potential. The study is based on a method of similar operators developed by A. G. Baskakov and his collaborators. This method allows us to reduce the study of the operator to one with a block-diagonal matrix. Asymptotic estimates of eigenvalues, eigenvectors, and spectral projections of a difference operator are obtained.