{"paper":{"title":"Asymptotics of determinants of discrete Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Alain Bourget, Tyler McMillen","submitted_at":"2016-09-14T03:47:35Z","abstract_excerpt":"We consider the asymptotics of the determinants of large discrete Schr\\\"odinger operators, i.e. \"discrete Laplacian $+$ diagonal\": \\[T_n(f) = -[\\delta_{j,j+1}+\\delta_{j+1,j}] + \\mbox{diag}\\left(f\\left(\\frac{1}{n}\\right), f\\left(\\frac{2}{n}\\right),\\dots, f\\left(\\frac{n}{n}\\right)\\right) \\] We extend a result of M. Kac, who found a formula for \\[\\lim_{n\\rightarrow\\infty} \\frac{\\det(T_n(f))}{G(f)^n} \\] in terms of the values of $f$, where $G(f)$ is a constant. We extend this result in two ways: First, we consider shifting the index: Let \\[T_n(f;\\varepsilon) = -[\\delta_{j,j+1}+\\delta_{j+1,j}] + \\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04125","kind":"arxiv","version":2},"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"}