{"paper":{"title":"P\\'olya's conjecture for thin products","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.SP","authors_text":"Xiang He, Zuoqin Wang","submitted_at":"2024-02-19T12:16:15Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb R^d$ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue $\\lambda_k(\\Omega)$ and its Neumann eigenvalue $\\mu_k(\\Omega)$ have the same leading asymptotics $w_k(\\Omega)=C(d,\\Omega)k^{2/d}$ as $k \\to \\infty$. G. P\\'olya conjectured in 1954 that each Dirichlet eigenvalue $\\lambda_k(\\Omega)$ is greater than $w_k(\\Omega)$, while each Neumann eigenvalue $\\mu_k(\\Omega)$ is no more than $w_k(\\Omega)$. In this paper we prove P\\'olya's conjecture for thin products, i.e. domains of the form $(a\\Omega_1) \\times \\Omega_2$, where $\\Omega_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.12093","kind":"arxiv","version":3},"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/2402.12093/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"}