{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:RLEFPNCEOIZ46AWBPZL7V652FV","short_pith_number":"pith:RLEFPNCE","schema_version":"1.0","canonical_sha256":"8ac857b4447233cf02c17e57fafbba2d4bf81f0318b29e9034df00b795873efc","source":{"kind":"arxiv","id":"2606.11677","version":1},"attestation_state":"computed","paper":{"title":"Feynman--Kac formula for the heat equation with a one-center point interaction in $d=3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Makoto Nakashima","submitted_at":"2026-06-10T05:41:17Z","abstract_excerpt":"We study Schr\\\"odinger operators with a one-center point interaction, formally defined by \\begin{align*} -\\Delta_\\alpha=-\\Delta+\\alpha\\,\\delta_0(\\cdot), \\end{align*} for $\\alpha\\in\\mathbb{R}$, and the associated heat equation \\begin{align} \\partial_t u=\\tfrac{1}{2}\\Delta_{\\alpha} u,\\quad u(0,x)=u_0(x)\\in C_c^{\\infty}(\\mathbb{R}^3\\setminus\\{0\\}).\\label{eq:HEapp} \\end{align} Here $\\Delta$ denotes the Laplacian (self-adjoint on $L^2(\\mathbb{R}^3)$) and $\\delta_x$ the Dirac measure at $x$. The operator $-\\Delta_\\alpha$ can be realized either as a self-adjoint extension of $-\\Delta|_{C_0^{\\infty}(\\"},"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":"2606.11677","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-06-10T05:41:17Z","cross_cats_sorted":[],"title_canon_sha256":"733df33e8df50f9a43450b2eb137078496f24f27b333e7ce31117732c51eccbd","abstract_canon_sha256":"51e910d78bf1f807c7dd4b8ebf120851a83511e61b5bb3df190212cd55b9a216"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:10:02.366380Z","signature_b64":"J/NPYeYuMgtCRHmcmbgxb3YsxXiQiwn+/dTKG7l2qs3tx/+LD081EN5ulokLhs4kx6WdGmEATvAZcJHV5dHdBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ac857b4447233cf02c17e57fafbba2d4bf81f0318b29e9034df00b795873efc","last_reissued_at":"2026-06-11T01:10:02.365591Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:10:02.365591Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Feynman--Kac formula for the heat equation with a one-center point interaction in $d=3$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Makoto Nakashima","submitted_at":"2026-06-10T05:41:17Z","abstract_excerpt":"We study Schr\\\"odinger operators with a one-center point interaction, formally defined by \\begin{align*} -\\Delta_\\alpha=-\\Delta+\\alpha\\,\\delta_0(\\cdot), \\end{align*} for $\\alpha\\in\\mathbb{R}$, and the associated heat equation \\begin{align} \\partial_t u=\\tfrac{1}{2}\\Delta_{\\alpha} u,\\quad u(0,x)=u_0(x)\\in C_c^{\\infty}(\\mathbb{R}^3\\setminus\\{0\\}).\\label{eq:HEapp} \\end{align} Here $\\Delta$ denotes the Laplacian (self-adjoint on $L^2(\\mathbb{R}^3)$) and $\\delta_x$ the Dirac measure at $x$. The operator $-\\Delta_\\alpha$ can be realized either as a self-adjoint extension of $-\\Delta|_{C_0^{\\infty}(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11677","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11677/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.11677","created_at":"2026-06-11T01:10:02.365713+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.11677v1","created_at":"2026-06-11T01:10:02.365713+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11677","created_at":"2026-06-11T01:10:02.365713+00:00"},{"alias_kind":"pith_short_12","alias_value":"RLEFPNCEOIZ4","created_at":"2026-06-11T01:10:02.365713+00:00"},{"alias_kind":"pith_short_16","alias_value":"RLEFPNCEOIZ46AWB","created_at":"2026-06-11T01:10:02.365713+00:00"},{"alias_kind":"pith_short_8","alias_value":"RLEFPNCE","created_at":"2026-06-11T01:10:02.365713+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.11178","citing_title":"A submartingale for the probability of avoiding the origin in one-point interaction ground-state diffusion: $d \\in \\{2,3\\}$","ref_index":30,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV","json":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV.json","graph_json":"https://pith.science/api/pith-number/RLEFPNCEOIZ46AWBPZL7V652FV/graph.json","events_json":"https://pith.science/api/pith-number/RLEFPNCEOIZ46AWBPZL7V652FV/events.json","paper":"https://pith.science/paper/RLEFPNCE"},"agent_actions":{"view_html":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV","download_json":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV.json","view_paper":"https://pith.science/paper/RLEFPNCE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.11677&json=true","fetch_graph":"https://pith.science/api/pith-number/RLEFPNCEOIZ46AWBPZL7V652FV/graph.json","fetch_events":"https://pith.science/api/pith-number/RLEFPNCEOIZ46AWBPZL7V652FV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV/action/storage_attestation","attest_author":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV/action/author_attestation","sign_citation":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV/action/citation_signature","submit_replication":"https://pith.science/pith/RLEFPNCEOIZ46AWBPZL7V652FV/action/replication_record"}},"created_at":"2026-06-11T01:10:02.365713+00:00","updated_at":"2026-06-11T01:10:02.365713+00:00"}