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 L ²-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 Lindelöf 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 ${\epsilon > 0}$.