Discrete Bethe-Sommerfeld Conjecture

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 Schrödinger 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.

10 pages

References in corpus (1)

• Bethe-Sommerfeld Conjecture

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