{"paper":{"title":"Sharp bound on the largest positive eigenvalue for one-dimensional Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Wencai Liu","submitted_at":"2017-09-17T06:17:04Z","abstract_excerpt":"Let $H=-D^2+V$ be a Schr\\\"odinger operator on $ L^2(\\mathbb{R})$, or on $ L^2(0,\\infty)$. Suppose the potential satisfies $\\limsup_{x\\to \\infty}|xV(x)|=a<\\infty$. We prove that $H$ admits no eigenvalue larger than $ \\frac{4a^2}{\\pi^2}$. For any positive $a$ and $\\lambda$ with $0<\\lambda< \\frac{4a^2}{\\pi^2}$, we construct potentials $V$ such that $\\limsup_{x\\to \\infty}|xV(x)|=a $ and the associated Sch\\\"rodinger operator $H=-D^2+V$ has eigenvalue $\\lambda$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05611","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"}