The essential spectrum of Schr\"{o}dinger operators on lattices

TL;DR

This paper applies the limit operators method to analyze the essential spectrum of various classes of discrete Schr"{o}dinger operators on lattices, demonstrating its effectiveness for both classical and complex potential scenarios.

Contribution

It extends the limit operators method to discrete Schr"{o}dinger operators, covering cases with oscillating, periodic, semi-periodic, discontinuous, and multi-particle potentials.

Findings

Effective description of the essential spectrum for various potential classes.

Demonstration of the limit operators method's applicability to discrete lattice problems.

Analysis of multi-particle Schr"{o}dinger operators on lattices.

Abstract

The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schr\"{o}dinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schr\"{o}dinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schr\"{o}dinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schr\"{o}dinger operators which describe the motion of two particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.