On the Spectrum of Multi-Frequency Quasiperiodic Schr\"odinger Operators with Large Coupling

TL;DR

This paper proves that under certain conditions, multi-frequency quasiperiodic Schrödinger operators with large coupling have spectra that form a single continuous interval, using a non-perturbative criterion related to positive Lyapunov exponents.

Contribution

It establishes a non-perturbative criterion for the spectrum to contain an interval, demonstrating the spectrum's structure for large coupling in quasiperiodic Schrödinger operators.

Findings

Spectrum consists of a single interval for large coupling.

Established a non-perturbative criterion for spectral intervals.

Proved the result for operators with positive Lyapunov exponent.

Abstract

We study multi-frequency quasiperiodic Schr\"{o}dinger operators on $\mathbb{Z} $. We prove that for a large real analytic potential satisfying certain restrictions the spectrum consists of a single interval. The result is a consequence of a criterion for the spectrum to contain an interval at a given location that we establish non-perturbatively in the regime of positive Lyapunov exponent.