{"paper":{"title":"Kernel estimates for Schr\\\"odinger type operators with unbounded diffusion and potential terms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abdelaziz Rhandi, Anna Canale, Cristian Tacelli","submitted_at":"2015-01-05T11:06:26Z","abstract_excerpt":"We prove that the heat kernel associated to the Schr\\\"odinger type operator $A:=(1+|x|^\\alpha)\\Delta-|x|^\\beta$ satisfies the estimate $$k(t,x,y)\\leq c_1e^{\\lambda_0t}e^{c_2t^{-b}}\\frac{(|x||y|)^{-\\frac{N-1}{2}-\\frac{\\beta-\\alpha}{4}}}{1+|y|^\\alpha} e^{-\\frac{2}{\\beta-\\alpha+2}|x|^{\\frac{\\beta-\\alpha+2}{2}}} e^{-\\frac{2}{\\beta-\\alpha+2}|y|^{\\frac{\\beta-\\alpha+2}{2}}} $$ for $t>0,|x|,|y|\\ge 1$, where $c_1,c_2$ are positive constants and $b=\\frac{\\beta-\\alpha+2}{\\beta+\\alpha-2}$ provided that $N>2,\\,\\alpha\\geq 2$ and $\\beta>\\alpha-2$. We also obtain an estimate of the eigenfunctions of $A$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00816","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"}