Discrete Bethe-Sommerfeld Conjecture

TL;DR

This paper proves a discrete version of the Bethe-Sommerfeld conjecture, showing the spectral structure of multi-dimensional discrete periodic Schrödinger operators with small potentials, including conditions for spectrum gaps.

Contribution

It establishes the spectral properties of discrete periodic Schrödinger operators, demonstrating the number of spectral intervals and conditions for gaps, extending the conjecture to discrete settings.

Findings

Spectra contain at most two intervals for small potentials.

Spectrum is a single interval if one period is odd.

Spectral gaps can occur when all periods are even.

Abstract

In this paper, we prove a discrete version of the Bethe-Sommerfeld conjecture. Namely, we show that the spectra of multi-dimensional discrete periodic Schr\"odinger operators on $\mathbb{Z}^d$ lattice with sufficiently small potentials contain at most two intervals. Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even.