This article is cited in 2 papers

On Spectral Properties of the Discrete Schrödinger Operator with Pure Imaginary Finite Potential

M. M. Faddeev

Saint-Petersburg State University

Abstract: In this paper, we consider the spectral properties of the discrete Schrödinger operator in the space of square integrable two-sided sequences with a pure imaginary potential of finite rank with zero mean value. We show that if such potentials are small, then the spectrum of the operator under study coincides with the spectrum of the unperturbed operator, and the operator itself is similar to a self-adjoint operator.

Keywords: discrete Schrödinger operator, spectral problem, $\mathscr{PT}$-symmetric potential, similarity to a self-adjoint operator.

UDC: 517.948.35

Received: 27.02.2008

DOI: 10.4213/mzm4670

 English version: Mathematical Notes, 2009, 85:3, 437–440

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