The essential spectrum of Schrodinger operators on lattices

The paper is devoted to the study of the essential spectrum of discrete Schrodinger operators on the lattice 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 Schrodinger operators, square-root Klein-Gordon operators and Dirac operators under quite weak assumptions on the behaviour of the magnetic and electric potential at infinity. The present paper aims at illustrating the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schrodinger operators: ( 1) operators with slowly oscillating at infinity potentials, ( 2) operators with periodic and semi-periodic potentials, ( 3) Schrodinger operators which are discrete quantum analogues of the acoustic propagators for waveguides, ( 4) operators with potentials having an infinite set of discontinuities and ( 5) three-particle Schrodinger operators which describe the motion of two particles around a heavy nuclei on the lattice Z(3).