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

We show that the spectrum of a discrete two-dimensional periodic Schrodinger 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 Kruger 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 almostperiodic operators.