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Abstract:We prove the positivity and continuity of the Lyapunov exponent for one-dimensional discrete Schrödinger operators with multi-frequency skew-shift potentials. For the operator $H_{\lambda,\omega} = \Delta + \lambda v(T_{\omega}^n(x,y))$ on $\ell^2(\mathbb{Z})$, where $T_{\omega}$ is a skew-shift on $\mathbb{T}^{d}\times\mathbb{T}^{d}$ $(d\geq1)$ and $v$ is a non-constant real-analytic function on $\mathbb{T}^{2d}$, we establish that for Diophantine frequency vectors $\omega$ and large coupling $\lambda \gg 1$, the Lyapunov exponent satisfies $L(\lambda,E) \geq c\log\lambda > 0$ uniformly in $E$ (with $c>0$), and is log-Hölder continuous in $E$. This work extends the known results of Lyapunov exponents--previously developed for one-frequency or simpler quasi-periodic models--to the genuinely multi-frequency skew-shift setting.

Submission history

From: Daxiong Piao [view email]
[v1] Mon, 22 Jun 2026 13:31:47 UTC (25 KB)