On the Genericity of the Spectrum Intervalization for Multi-Frequency Quasiperiodic Schrödinger Operators

Abstract

This paper proves a genericity conjecture by Goldstein, Schlag, and Voda[Invent. Math.\textbf{217}(2019)] for multi-frequency quasiperiodic Schrödinger 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.

arXiv commentThis manuscript is withdrawn due to substantial mathematical errors in the formulation and proof of the main theorem. In particular, the key quantitative estimates used in the argument do not hold as stated. Consequently, the claimed generic interval-spectrum result has not been established