Quantitative quantum ergodicity and the nodal domains of Maass-Hecke 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 $O_{\epsilon }\left(t_{\varphi }^{3/8+\epsilon }\right)$ 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 $O_{\epsilon }\left(t_{\varphi }^{5/12+\epsilon }\right)$ 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 $t_{\varphi }^{1/12-\epsilon }$ for any $\epsilon >0$, under the assumption that the Lindel{ö}f 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 $t_{\varphi }^{1/8-\epsilon }$ for any $\epsilon >0$.

37 pages, added details, and fixed minor errors

References in corpus (2)

• Nodal domains of Maass forms I
• Quantum unique ergodicity for SL_2(Z)\H

Cited by in corpus (3)

• Equidistribution of Eisenstein series on geodesic segments
• Arithmetic, zeros, and nodal domains on the sphere
• On the nodal lines of Eisenstein series on Schottky surfaces