On the Genericity of the Spectrum Intervalization for Multi-Frequency Quasiperiodic Schrödinger Operators
Daxiong Piao
TL;DR
The paper addresses whether the spectrum of the multi-frequency quasiperiodic Schrödinger operator $H(x)\psi(n) = -\psi(n+1) - \psi(n-1) + \lambda V(x + n\omega)\psi(n)$ is an interval for generic potentials. It proves that for real trigonometric polynomial potentials $V(x;\mathbf{c})$ with coefficient vector $\mathbf{c}\in\mathbb{R}^N$, the potential belongs to class $\mathfrak{G}$ for almost every $\mathbf{c}$, and thus for large $\lambda$, the spectrum is a single interval. The proof weaves parametric transversality and Cartan-type estimates to show Morseness, unique extrema, and non-degeneracy hold generically; then existing results translate this to spectral-interval behavior. This establishes the Goldstein–Schlag–Voda genericity conjecture in full-measure sense, extending intuition from Chulaevsky–Sinai and providing a geometric/topological framework for spectral genericity in the multi-frequency setting.
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.