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

This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice Zn which are discrete analogs of the Schrödinger, 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 the so-called pseudodifference operators (i.e., pseudodifferential operators on the group Zn with analytic symbols, as developed in the paper [29]), and the limit operators method (see [34] and the references cited there). 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 Schrödinger operators on Zn, Dirac operators on Z, and square root Klein-Gordon operators on Zn.