{"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"}