Quantitative Quantum Ergodicity and the Nodal Domains of Hecke-Maass Cusp Forms

Abstract

We prove a quantitative statement of the quantum ergodicity for Hecke-Maass cusp forms on the modular surface. As an application of our result, along a density 1 subsequence of even Hecke-Maass cusp forms, we obtain a sharp lower bound for the L2-norm of the restriction to a fixed compact geodesic segment of eta = {iy : y > 0} subset of H. We also obtain an upper bound of O-epsilon (t(phi)(3/8+epsilon)) for the L-infinity norm along a density 1 subsequence of Hecke-Maass cusp forms; for such forms, this is an improvement over the upper bound of O-epsilon (t(phi)(5/12+epsilon)) given by Iwaniec and Sarnak. In a recent work of Ghosh, Reznikov, and Sarnak, the authors proved for all even Hecke-Maass forms that the number of nodal domains, which intersect a geodesic segment of eta, grows faster than t(phi)(1/12-epsilon) for any epsilon > 0, under the assumption that the Lindelof Hypothesis is true and that the geodesic segment is long enough. Upon removing a density zero subset of even Hecke-Maass forms, we prove without making any assumptions that the number of nodal domains grows faster than t(phi)(1/8+epsilon) for any epsilon > 0