Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials

  • Mark Embree

    Virginia Tech, Blacksburg, USA
  • Jake Fillman

    Virginia Tech, Blacksburg, USA
Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials cover

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Abstract

We show that the spectrum of a discrete two-dimensional periodic Schrödinger 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 Krüger 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.

Cite this article

Mark Embree, Jake Fillman, Spectra of discrete two-dimensional periodic Schrödinger operators with small potentials. J. Spectr. Theory 9 (2019), no. 3, pp. 1063–1087

DOI 10.4171/JST/271