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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"},"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":"1005.2690","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2010-05-15T17:26:49Z","cross_cats_sorted":[],"title_canon_sha256":"886a7434ad900d7d147b2feaa4a7ec678a5301412202db9e6890a2b6bacf34be","abstract_canon_sha256":"4291a0eb2a85fcafa44b0dd0a43abe4997be307dca0479f0cb42210b6a62fe39"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:46.279184Z","signature_b64":"8+tdzmjQhhywiMWx1abvREpw9BKtS9SY0qmA/aY6elHOMNJkRL6tw2uj2QCS2AEVL4EusLC+ZN13fgUYFLwzDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69842582759a079cc33aed403afc38e6152fa45881b590dabd54d6bc3a0805b6","last_reissued_at":"2026-05-18T04:40:46.278431Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:46.278431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the spectral estimates for Schr\\\"odinger type operators. 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