2014/01/08 by Suresh Eswarathasan, Eswarathasan, Suresh · 1 citation
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1401.1710
openalex publication_date 2014/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (M,g) be a compact Riemannian surface. Consider a family of L2 normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form -hj2Δg ϕhj = ϕhj, whose eigenvalues satisfy h hj-1 ∈ (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], C1 h1/2 ≤ Eh [ | ∫γ u d σ| ] ≤ C2 h1/2 . This result contrasts the deterministic O(1) upperbounds obtained by Chen-Sogge \citeCS, Reznikov \citeRez, and Zelditch \citeZel. 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.