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

Abstract

We study discrete Schrödinger operators on \(\ell^2(\mathbb Z)\) generated by the vector-frequency two-step skew shift \(T_ω(x,y)=(x+ω,x+y)\), with an arbitrary nonconstant real-analytic sampling function on \(\mathbb T^d\times\mathbb T^d\). For frequencies in a fixed vector Diophantine class and sufficiently large coupling, we prove stretched-exponential large-deviation estimates at every sufficiently large integer scale, uniformly in the real energy and in the frequency. We also obtain positivity of the Lyapunov exponent for all real energies and a quantitative local continuity estimate in the energy. The proof addresses the simultaneous losses caused by the \(n^{-1}\) complex width induced by the shear, vector Diophantine small divisors in quadratic Weyl sums, and several-variable subharmonic concentration. The main ingredients are a vector Weyl estimate with Poisson regularization, a short-block large-deviation estimate on anisotropic shrinking strips, a several-variable boosting argument, and an all-offset avalanche comparison followed by strong induction.