Exponential decay of the size of spectral gaps for quasiperiodic Schr\"odinger operators

TL;DR

This paper proves that spectral gaps in quasiperiodic Schrödinger operators decay exponentially with their label, refining previous bounds, using non-perturbative methods based on reducibility estimates, and shows the spectrum is 1/2-homogeneous.

Contribution

It establishes exponential decay of spectral gaps for quasiperiodic Schrödinger operators using non-perturbative techniques, improving upon prior subexponential bounds.

Findings

Spectral gaps decay exponentially with their label.

Spectrum is 1/2-homogeneous under the given conditions.

Refinement of previous subexponential bounds.

Abstract

In the following we are interested in the spectral gaps of discrete quasiperiodic Schr\"odinger operators when the frequency is Diophantine, the potential is analytic, and in the subcritical regime. The gap-labelling theorem asserts in this context that each gap has constant rotation number, labeled by some integer. We prove that the size of these gaps decays exponentially fast with respect to their label. This refines a subexponential bound obtained previously by Sana Ben Hadj Amor. Contrary to her approach, which is based on KAM methods, the arguments in the present paper are non-perturbative, and use quantitative reducibility estimates obtained by Artur Avila and Svetlana Jitomirskaya. As a corollary of our result, we show that under the previous assumptions, the spectrum is 1/2-homogeneous.