On the Genericity of the Spectrum Intervalization for Multi-Frequency Quasiperiodic Schr\"{o}dinger Operators

TL;DR

This paper proves that for most multi-frequency quasiperiodic Schr"odinger operators with trigonometric polynomial potentials, the spectrum is a single interval under strong coupling, confirming a long-standing conjecture.

Contribution

It establishes that the spectrum forms a single interval for generic potentials, extending previous results to a full measure setting using advanced mathematical tools.

Findings

Spectrum is a single interval for almost all coefficients under strong coupling.

Confirms the genericity conjecture by Goldstein, Schlag, and Voda.

Extends previous results to a full measure setting.

Abstract

This paper proves a genericity conjecture by Goldstein, Schlag, and Voda[Invent. Math.\textbf{217}(2019)] for multi-frequency quasiperiodic Schr\"{o}dinger operators. Specifically, we show that for almost all coefficients of real trigonometric polynomial potentials, the spectrum forms a single interval under strong coupling conditions. This confirms a long-standing intuition by Chulaevky and Sinai[Comm.Math.Phys.\textbf{125}(1989)] that the spectrum typically intervals for generic potentials, and extends the existence results of Goldstein et al. to a full measure setting. Our proof relies on tools from differential topology, measure theory, and analytic function theory.