Spectra of Discrete Two-Dimensional Periodic Schr\"odinger Operators with Small Potentials

TL;DR

This paper proves that the spectrum of small-potential 2D discrete periodic Schrödinger operators on a square lattice is mostly a single interval, with at most two, and gaps only at zero energy, refining previous results and relating to the Bethe–Sommerfeld conjecture.

Contribution

It establishes the spectral structure of small-potential 2D discrete Schrödinger operators, showing the spectrum is an interval or at most two, and identifies conditions for gap openings, advancing understanding of spectral properties.

Findings

Spectrum is an interval for odd periods in at least one dimension.

Spectrum may consist of at most two intervals.

Gaps can only open at zero energy.

Abstract

We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that the spectrum may consist of at most two intervals and that a gap may only open at energy zero. This sharpens several results of Kr\"uger and may be thought of as a discrete version of the Bethe--Sommerfeld conjecture. We also describe an application to the study of two-dimensional almost-periodic operators.