{"paper":{"title":"Level statistics of one-dimensional Schr\\\"odinger operators with random decaying potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Fumihiko Nakano, Shinichi Kotani","submitted_at":"2012-10-16T00:36:30Z","abstract_excerpt":"We study the level statistics of one-dimensional Schr\\\"odinger operator with random potential decaying like $x^{-\\alpha}$ at infinity. We consider the point process $\\xi_L$ consisting of the rescaled eigenvalues and show that : (i)(ac spectrum case) for $\\alpha > \\frac 12$, $\\xi_L$ converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian. (ii)(critical case) for $\\alpha = \\frac 12$, $\\xi_L$ converges to the limit of the circular $\\beta$-ensemble."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4224","kind":"arxiv","version":3},"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"}