{"paper":{"title":"Weyl laws for exponentially small singular values of the $\\overline{\\partial}$ operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CV"],"primary_cat":"math.SP","authors_text":"Johannes Sj\\\"ostrand, Martin Vogel, Michael Hitrik","submitted_at":"2025-05-12T07:25:34Z","abstract_excerpt":"We study the number of exponentially small singular values of the semiclassical $\\overline{\\partial}$ operator on exponentially weighted $L^2$ spaces on a compact Riemann surface. Accurate upper and lower bounds on the number of such singular values are established in terms of auxiliary notions of upper and lower bound weights. Assuming that the Laplacian of the exponential weight changes sign along a curve, we construct optimal such weights by solving a free boundary problem, which yields Weyl asymptotics for the counting function of the singular values in an interval of the form $[0,\\mathrm{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.07292","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/2505.07292/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"}