Eswarathasan, Suresh
2016.
Expected values of eigenfunction periods.
Journal of Geometric Analysis
26
, pp. 360-377.
10.1007/s12220-014-9554-6
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Abstract
Let (M,g) be a compact Riemannian surface. Consider a family of L2 normalized Laplace–Beltrami eigenfunctions, written in the semiclassical form −h 2 j Δgφhj=φhj, whose eigenvalues satisfy hh −1 j ∈(1,1+hD] for D>0 a large enough constant. Let Ph be a uniform probability measure on the L2 unit-sphere Sh of this cluster of eigenfunctions and take u∈Sh. Given a closed curve γ⊂M, there exists C1(γ,M),C2(γ,M)>0 and h0>0 such that for all h∈(0,h0], C1h1/2≤Eh[|∫γudσ|]≤C2h1/2. This result contrasts the previous deterministic O(1) upperbounds obtained by Chen–Sogge, Reznikov, and Zelditch. Furthermore, we treat the higher dimensional cases and compute large deviation estimates. Under a measure zero assumption on the periodic geodesics in S∗M, we can consider windows of small width D=1 and establish a O(h1/2) estimate. Lastly, we treat probabilistic Lq restriction bounds along curves.
| Item Type: | Article |
|---|---|
| Date Type: | Publication |
| Status: | Published |
| Schools: | Schools > Mathematics |
| Subjects: | Q Science > QA Mathematics |
| Publisher: | Springer Verlag (Germany) |
| ISSN: | 1050-6926 |
| Date of First Compliant Deposit: | 30 March 2016 |
| Last Modified: | 03 Dec 2024 09:30 |
| URI: | https://orca.cardiff.ac.uk/id/eprint/86428 |
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