On the spectra of separable 2D almost Mathieu operators

TL;DR

This paper proves that for small couplings, the spectrum of separable 2D almost Mathieu operators with Diophantine frequencies is an interval, extending previous results and using advanced spectral analysis techniques.

Contribution

It establishes that small coupling parameters lead to continuous spectra in 2D almost Mathieu operators, generalizing to multidimensional cases with analytic potentials.

Findings

Spectrum is an interval for small couplings and Diophantine frequencies.

Extends results to multidimensional quasiperiodic operators.

Uses the Newhouse Gap Lemma to analyze spectrum thickness.

Abstract

We consider separable 2D discrete Schr\"odinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schr\"odinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.