On localization and the spectrum of multi-frequency quasi-periodic operators

TL;DR

This paper investigates multi-frequency quasi-periodic Schrödinger operators, establishing exponential localization and eigenvalue separation in the positive Lyapunov exponent regime, and explores the spectral structure for certain potentials.

Contribution

It combines advanced elimination techniques to prove localization and spectral properties for general analytic potentials, advancing understanding of multi-frequency operators.

Findings

Proves exponential finite-volume localization.

Establishes eigenvalue separation.

Shows spectrum can be a single interval for specific potentials.

Abstract

We study multi-frequency quasi-periodic Schr\"odinger operators on $\mathbb{Z}$ in the regime of positive Lyapunov exponent and for general analytic potentials. Combining Bourgain's semi-algebraic elimination of multiple resonances with the method of elimination of double resonances via resultants, we establish exponential finite-volume localization as well as the separation between the eigenvalues. In a follow-up paper we develop the method further to show that for potentials given by large generic trigonometric polynomials the spectrum consists of a single interval, as conjectured by Chulaevski and Sinai.