Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics

TL;DR

This paper analyzes the essential spectra and exponential decay of eigenfunctions for lattice operators in quantum mechanics, using pseudodifference operators and limit operators to describe spectral locations and decay estimates.

Contribution

It introduces a novel approach combining pseudodifference operators and limit operators to analyze spectra and eigenfunction decay in discrete quantum lattice operators.

Findings

Describes the essential spectra of lattice Schrödinger, Dirac, and Klein-Gordon operators.

Provides exponential decay estimates for eigenfunctions of these operators.

Offers a unified framework for spectral analysis of discrete quantum operators.

Abstract

This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice Z^n which are discrete analogs of the Schr\"{o}dinger, Dirac and square-root Klein-Gordon operators. Our investigation of the essential spectra and the exponential decay of eigenfunctions of the discrete spectra is based on the calculus of so-called pseudodifference operators (i.e., pseudodifferential operators on the group Z^n) with analytic symbols and on the limit operators method. We obtain a description of the location of the essential spectra and estimates of the eigenfunctions of the discrete spectra of the main lattice operators of quantum mechanics, namely: matrix Schr\"{o}dinger operators on Z^n, Dirac operators on Z^3, and square root Klein-Gordon operators on Z^n.