{"paper":{"title":"To $1/2$-logconcavity and beyond: Geometric properties of Dirichlet eigenfunctions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jin Sun, Kui Wang, Lei Qin","submitted_at":"2026-06-02T14:07:49Z","abstract_excerpt":"We prove that, on a bounded open convex domain $\\Omega\\subset\\mathbb{R}^n$, the first Dirichlet eigenfunction of the Laplacian or the Ornstein--Uhlenbeck operator is $\\alpha$-logconcave for every $\\alpha\\in(0,1/2]$. This extends the recent $1/2$-logconcavity theorem of Crasta--Fragal\\`{a} for the Laplacian to the weighted Gaussian setting and, simultaneously, to a broader range of exponents. More precisely, if $u$ denotes the first eigenfunction normalized by $\\|u\\|_\\infty=1$, then for every $\\alpha\\in(0,1/2]$, the function $-\\bigl(-\\log(\\kappa u(x))\\bigr)^{\\alpha}$ is concave in $\\Omega$ prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03684","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/2606.03684/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"}