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We also obtain an estimate of the eigenfunctions of $A$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.00816","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-01-05T11:06:26Z","cross_cats_sorted":[],"title_canon_sha256":"27ac7257b711aa9b3ce799100d9301de1e806f7b1bc287026286fb1c8e9a7c77","abstract_canon_sha256":"6ed8d94ab5fbce2334b323ede0e80158ccbaea67afa1b37e0c9b2b4f79a614b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:05.068362Z","signature_b64":"26K3vWIvffqDhOZbwPKDj9WCd8r1Vh+D7Bdq8+X3+c5wphEj6mqWog3pcoeI0KWyPefuKvypJPAs08ZF+h6xBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f7d7d8589da8c4c81d7046a4adc7e7e9f408c035ac663266065d97773aa25c47","last_reissued_at":"2026-05-18T00:57:05.067881Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:05.067881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\beta>\\alpha-2$. 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