{"paper":{"title":"Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic","license":"","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"David Damanik (Caltech), Rowan Killip (UCLA)","submitted_at":"2004-02-25T06:24:55Z","abstract_excerpt":"We prove absence of absolutely continuous spectrum for discrete one-dimensional Schr\\\"odinger operators on the whole line with certain ergodic potentials, $V_\\omega(n) = f(T^n(\\omega))$, where $T$ is an ergodic transformation acting on a space $\\Omega$ and $f: \\Omega \\to \\R$. The key hypothesis, however, is that $f$ is discontinuous. In particular, we are able to settle a conjecture of Aubry and Jitomirskaya--Mandel'shtam regarding potentials generated by irrational rotations on the torus.\n  The proof relies on a theorem of Kotani, which shows that non-deterministic potentials give rise to ope"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0402070","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"}