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 \({\eta=\{iy : y > 0\} \subset {\mathbb{H}}}\). We also obtain an upper bound of \({O_\epsilon\left(t_\phi^{3/8+\epsilon} \right)}\) for the \({L^\infty}\) norm along a density 1 subsequence of Hecke–Maass cusp forms; for such forms, this is an improvement over the upper bound of \({O_\epsilon\left(t_\phi^{5/12+\epsilon} \right)}\) 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 \({\epsilon > 0}\).