Polynomial Spectral Gap Estimates for Finitely Differentiable Quasi-Periodic Schrödinger Operators

Abstract

We prove polynomial upper bounds on spectral gaps of quasi-periodic Schrödinger operators with small $C^{k}$ potentials and Diophantine frequencies. For sufficiently large $k$, the decay exponent is $k-34τ$, where $τ$ is the Diophantine exponent. We construct examples with lower bounds along infinitely many gap labels, showing that the leading coefficient $1$ of $k$ is asymptotically optimal. The proof uses quantitative reducibility and the Moser-Pöschel argument.