{"paper":{"title":"On spectral estimates for two-dimensional Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"A. Laptev, M. Solomyak","submitted_at":"2012-01-15T09:43:22Z","abstract_excerpt":"For a two-dimensional Schr\\\"odinger operator $H_{\\alpha V}=-\\Delta-\\alpha V,\\ V\\ge 0,$ we study the behavior of the number $N_-(H_{\\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\\alpha$ tends to infinity. A wide class of potentials is described, for which $N_-(H_{\\alpha V})$ has the semi-classical behavior, i.e., $N_-(H_{\\alpha V})=O(\\alpha)$. For the potentials from this class, the necessary and sufficient condition is found for the validity of the Weyl asymptotic law."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3074","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"}