{"paper":{"title":"Expected values of eigenfunction periods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","math.SP"],"primary_cat":"math.AP","authors_text":"Suresh Eswarathasan","submitted_at":"2014-01-08T14:46:27Z","abstract_excerpt":"Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2\\Delta_g \\phi_{h_j} = \\phi_{h_j}$, whose eigenvalues satisfy $h h_j^{-1} \\in (1, 1 + hD]$ for $D>0$ a large enough constant. Let $\\mathbf{P}_h$ be a uniform probability measure on the $L^2$ unit-sphere $S_h$ of this cluster of eigenfunctions and take $u \\in S_h$. Given a closed curve $\\gamma \\subset M$, there exists $C_{1}(\\gamma, M), C_{2}(\\gamma, M) > 0$ and $h_0>0$ such that for all $h \\in (0, h_0],$ \\begin{equation*}\n  C_1 h^{1/2} \\leq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.1710","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":""},"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"}