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 at least one of the periods is odd, and can have a gap whenever all periods are even.
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Han, R., Jitomirskaya, S. Discrete Bethe–Sommerfeld Conjecture. Commun. Math. Phys. 361, 205–216 (2018). https://doi.org/10.1007/s00220-018-3141-9
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DOI: https://doi.org/10.1007/s00220-018-3141-9