{"paper":{"title":"Weyl sums and the Lyapunov exponent for the skew-shift Schr\\\"odinger cocycle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Marius Lemm, Rui Han, Wilhelm Schlag","submitted_at":"2018-06-30T22:13:09Z","abstract_excerpt":"We study the one-dimensional discrete Schr\\\"odinger operator with the skew-shift potential $2\\lambda\\cos\\left(2\\pi \\left(\\binom{j}{2} \\omega+jy+x\\right)\\right)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $\\lambda>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(\\lambda)$ at small $\\lambda$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(\\lambda)$ is fully consiste"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.00233","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}