{"paper":{"title":"Asymptotic behavior of $L^2$-normalized eigenfunctions of the Laplace-Beltrami operator on a closed Riemannian manifold","license":"","headline":"","cross_cats":["math.AP"],"primary_cat":"math.SP","authors_text":"Bin Xu","submitted_at":"2005-09-04T11:50:45Z","abstract_excerpt":"Let $e(x,y,\\l)$ be the spectral function and ${\\chi}_\\l$ the unit band spectral projection operator, with respect to the Laplace-Beltrami operator $\\D_M$ on a closed Riemannian manifold $M$. We firstly review the one-term asymptotic formula of $e(x,x,\\l)$ as $\\l\\to\\infty$ by H{\\\" o}rmander (1968) and the one of $\\p^\\al_x\\p^\\bt_y e(x,y,\\l)|_{x=y}$ as $\\l\\to\\infty$ in a geodesic normal coordinate chart by the author (2004) and the sharp asymptotic estimates from above of the mapping norm $\\|\\chi_\\l\\|_{L_2\\to L_p}$ ($2\\leq p\\leq\\infty$) by Sogge (1988 $&$ 1989) and of the mapping norm $\\|\\chi_\\l\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0509061","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/math/0509061/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"}