{"paper":{"title":"On a class of spectral problems on the half-line and their applications to multi-dimensional problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Michael Solomyak","submitted_at":"2012-03-06T10:19:42Z","abstract_excerpt":"A survey of estimates on the number $N_-(\\BM_{\\a G})$ of negative eigenvalues (bound states) of the Sturm-Liouville operator $\\BM_{\\a G}u=-u\"-\\a G$ on the half-line, as depending on the properties of the function $G$ and the value of the coupling parameter $\\a>0$. The central result is \\thmref{S1/2a} giving a sharp sufficient condition for the semi-classical behavior $N_-(\\BM_{\\a G})=O(\\a^{1/2})$, and the necessary and sufficient conditions for a \"super-classical\" growth rate $N_-(\\BM_{\\a G})=O(\\a^q)$ with any given $q>1/2$. Similar results for the problem on the whole $\\R$ are also presented."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1156","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"}