{"paper":{"title":"Pointwise estimates for the fundamental solutions of higher order Schr\\\"{o}dinger equations in odd dimensions II: high dimensional case","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Han Cheng, Quan Zheng, Shanlin Huang, Tianxiao Huang","submitted_at":"2024-08-28T06:13:04Z","abstract_excerpt":"In this paper, for any odd $n$ and any integer $m\\geq1$ with $n>4m$, we study the fundamental solution of the higher order Schr\\\"{o}dinger equation \\begin{equation*} \\mathrm{i}\\partial_tu(x,t)=((-\\Delta)^m+V(x))u(x,t),\\quad t\\in \\mathbb{R},\\,\\,x\\in \\mathbb{R}^n, \\end{equation*} where $V$ is a real-valued $C^{\\frac{n+1}{2}-2m}$ potential with certain decay. Let $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum space of $H=(-\\Delta)^m+V$, and assume that $H$ has no positive embedded eigenvalue. Our main result says that $e^{-\\mathrm{i}tH}P_{ac}(H)$ has integral kernel $K("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.00117","kind":"arxiv","version":2},"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/2409.00117/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"}