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 Zd 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.
Publication Source (Journal or Book title)
Communications in Mathematical Physics
Han, R., & Jitomirskaya, S. (2018). Discrete Bethe–Sommerfeld Conjecture. Communications in Mathematical Physics, 361 (1), 205-216. https://doi.org/10.1007/s00220-018-3141-9