Discrete Bethe-Sommerfeld Conjecture

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.