{"paper":{"title":"Low-lying eigenvalues of semiclassical Schr\\\"odinger operator with degenerate wells","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Jean-Francois Bony, Nicolas Popoff","submitted_at":"2018-02-08T14:39:48Z","abstract_excerpt":"In this article, we consider the semiclassical Schr\\\"odinger operator $P = - h^{2} \\Delta + V$ in $\\mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $\\lambda_{k} ( P )$ as $h \\to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$ for $\\lambda_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) = \\{ 0 \\}$, we are able to control the eigenvalues $\\lambda_{k} ( P )$ for monotonous potentials by a quantity linked to an interval $I_{h}$, determined by an implicit relation involving $V$ and $h$. Next, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.02882","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"}