Positivity and log-Hölder Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators

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 $H_{λ,ω} = Δ+ λv(T_ω^n(x,y))$ on $\ell^2(\mathbb{Z})$, where $T_ω$ 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 $ω$ and large coupling $λ\gg 1$, the Lyapunov exponent satisfies $L(λ,E) \geq c\logλ> 0$ 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.