The essential spectrum of Schrödinger operators on lattices

Abstract

The paper is devoted to the study of the essential spectrum of discrete Schrödinger 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 Schrödinger 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 Schrödinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schrödinger 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 Schrödinger operators which describe the motion of two particles around a heavy nuclei on the lattice Z 3 .

Essential spectrum of difference operators on periodic metric spaces

“…Note that in the papers [9] and [10] we used formula (4) to give a description of the essential spectrum of Schrödinger operators on graphs periodic with respect to the action of the group Z n . In particular, we described the essential spectrum of discrete Schrödinger operators on hexagonal and zigzag graphs, important in the theory of nanostructures, as well as the essential spectrum of multiparticle problems on Z n -periodic graphs.…”

“…(Note that the Roe's Fredholm criterion does not imply formula (1), and so the present paper is the first to obtain (1) for the situation in question.) Formula (1) was applied to the essential spectrum of electromagnetic Schrödinger operators on the lattice Z n in [9] and to the essential spectrum of Schrödinger operators on combinatorial graphs periodic with respect to the group Z n in [10]. In contrast to these papers, here we consider operators on discrete structures periodic with respect to the action of noncommutative discrete groups.…”

Wiener algebra of operators on the lattice (μ ℤ)ndepending on the small parameter μ>0

“…In the [7,8,9] we studied the exponential estimates of solutions of the so-called pseudodifference equations which are discrete analogue of pseudodifferential equations on R n with applications to the discrete models of Schro¨dinger, Dirac and Klein-Gordon operators. Note that the discrete models of the quantum mechanics are intensively studied (see, for instance, [1,4,5,11] and references given therein).…”

“…PROPOSITION 21 ([6, p. 239], [7,8]) Let È 2 SO(R n ) be a real valued function. Then: Proof Note that for every 40 there exists 0 40 such that dist(x, (Z) n )5 for every point x 2 R n and every 2 (0, 0 ).…”

Fredholm Properties of Band-dominated Operators on Periodic Discrete Structures

“…In [34], this formula is applied to study electro-magnetic Schrödinger operators on the lattice Z n , and in [35] a generalization to Z n -periodic graphs is derived. A similar formula for essential spectra of perturbed pseudodifferential operators on R n were obtained in [32] and applied for investigation of location of essential spectra of electro-magnetic Schrödinger operators, square-root Klein-Gordon, and Dirac operators under general assumptions with respect to the behavior of magnetic and electric potentials at infinity.…”