54 research outputs found
Metal-insulator transition for the almost Mathieu operator
We prove that for Diophantine \om and almost every \th, the almost Mathieu
operator, (H_{\omega,\lambda,\theta}\Psi)(n)=\Psi(n+1) + \Psi(n-1) +
\lambda\cos 2\pi(\omega n +\theta)\Psi(n), exhibits localization for \lambda >
2 and purely absolutely continuous spectrum for \lambda < 2. This completes the
proof of (a correct version of) the Aubry-Andr\'e conjecture.Comment: 17 pages, published versio
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 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.Comment: 10 page
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