{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"c12de272d3b751bb186cf65f7eec081d16b0f0f81bf13192b92224db13e5baab","created":"2026-10-03T07:18:06Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"a89ccd7329102589b1ea87dd47da48ac840a9098bd338ccf084fe4d82e5050f5","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"mathph.quantum-chaos.que-negative-curvature","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"On a curved surface where every classical orbit is chaotic, every high-frequency standing wave should spread evenly over the surface. This is proven only for special surfaces with number-theoretic symmetry.","posed_since":"1994","precise":"Compact Riemannian surface $M$ of variable negative curvature, eigenfunctions $-\\operatorname{Lap} \\phi_{j} = \\lambda_{j} \\phi_{j}$ with $||\\phi_{j}||_{2} = 1$. Prove or disprove that $|\\phi_{j}|^{2} \\,\\mathrm{dvol} \\to \\mathrm{dvol}/\\operatorname{vol}(M)$ weakly for the whole sequence (Rudnick-Sarnak conjecture). Answer: yes or no with proof.","problem_ref":null,"references":"","settled_by":"A proof for all negatively curved surfaces, or a counterexample sequence of eigenfunctions with a non-uniform limit.","status_note":"Proven for Hecke eigenfunctions on arithmetic surfaces (Lindenstrauss, 2006; Soundararajan, 2010); density-one subsequences equidistribute by quantum ergodicity; for all negatively curved surfaces every semiclassical limit has full support (Dyatlov and Jin, 2018; Dyatlov, Jin and Nonnenmacher, 2022); Hassell's 2010 failure of QUE is for stadium billiards, which are not negatively curved surfaces.","title":"Quantum unique ergodicity on negatively curved 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