Upper bounds on the spectral gaps of quasi-periodic Schr\"odinger operators with Liouville frequencies

TL;DR

This paper demonstrates that spectral gaps in weakly coupled quasi-periodic Schrödinger operators with Liouville frequencies decay exponentially, leading to the spectrum's homogeneity.

Contribution

It establishes exponential decay bounds for spectral gaps in operators with Liouville frequencies, a novel result in spectral theory.

Findings

Spectral gaps decay exponentially with weak coupling.

Spectrum is homogeneous for these operators.

Provides new bounds on spectral gap sizes.

Abstract

We prove that the size of the spectral gaps of weakly coupled quasi-periodic Schr\"odinger operators with Liouville frequencies decays exponentially. As an application, we obtain the homogeneity of the spectrum.