{"paper":{"title":"Optimal Covariance Estimates for Schr\\\"odinger Semigroups with White Noise in $d=1,2$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Pierre Yves Gaudreau Lamarre, Youssef Djellouli","submitted_at":"2026-07-19T19:49:47Z","abstract_excerpt":"For $d\\in\\{1,2\\}$, let $H=-\\frac{1}{2}\\Delta + V +\\xi$ be the random Schr\\\"odinger operator on $L^2(\\mathbb{R}^d)$ where $\\xi$ is a standard Gaussian white noise and $V$ is a deterministic potential with power-law growth at infinity. Using a Feynman-Kac formula for the trace of the Schr\\\"odinger semigroup, we give optimal asymptotic upper and lower bounds on the covariance of $\\mathrm{Tr}[e^{-sH}]$ and $\\mathrm{Tr}[e^{-tH}]$ as $s,t\\to0$ through estimates on Brownian bridge local times. These estimates are a significant improvement on previous bounds in the case $d=1$ and are the first of thei"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.17393","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/2607.17393/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"}