Expected Values of Eigenfunction Periods

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.