Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators
Abstract
We study discrete Schrödinger operators on ℓ2(ℤ) generated by the vector-frequency two-step skew shift Tω(x,y)=(x+ω,x+y), with an arbitrary nonconstant real-analytic sampling function on 𝕋d×𝕋d. 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.
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BibTeX
@misc{wang2026positivity,
title = {{Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators}},
author = {Chao Wang and Yuanyuan Peng and Daxiong Piao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.23316},
eprint = {2606.23316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23316},
}
RIS
TY - UNPB
TI - Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators
AU - Chao Wang
AU - Yuanyuan Peng
AU - Daxiong Piao
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2606.23316
UR - https://arxiv.org/abs/2606.23316
AB - We study discrete Schrödinger operators on ℓ^2(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 𝕋^d×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.
ER -