{"paper":{"title":"On the spectral estimates for Schr\\\"odinger type operators. The case of small local dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Grigori Rozenblum, Michael Solomyak","submitted_at":"2010-05-15T17:26:49Z","abstract_excerpt":"The behavior of the discrete spectrum of the Schr\\\"odinger operator $-\\D - V$, in quite a general setting, up to a large extent is determined by the behavior of the corresponding heat kernel $P(t;x,y)$ as $t\\to 0$ and $t\\to\\infty$. If this behavior is powerlike, i.e., \\[\\|P(t;\\cdot,\\cdot)\\|_{L^\\infty}=O(t^{-\\delta/2}),\\ t\\to 0;\\qquad \\|P(t;\\cdot,\\cdot)\\|_{L^\\infty}=O(t^{-D/2}),\\ t\\to\\infty,\\] then it is natural to call the exponents $\\delta,D$ \"{\\it the local dimension}\" and \"{\\it the dimension at infinity}\" respectively. The character of spectral estimates depends on the relation between thes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.2690","kind":"arxiv","version":3},"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"}