Quantitative quantum ergodicity and the nodal domains of Maass-Hecke cusp forms

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 $O_{ϵ}\left(t_{\varphi }^{3\mathrm{/}8+ϵ}\right)$ for the $L^{\mathrm{\infty }}$ norm along a density $1$ subsequence of Hecke–Maass cusp forms; for such forms, this is an improvement over the upper bound of $O_{ϵ}\left(t_{\varphi }^{5\mathrm{/}12+ϵ}\right)$ 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 $t_{\varphi }^{1\mathrm{/}12-ϵ}$ for any $ϵ>0$, under the assumption that the Lindel{"o}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 $t_{\varphi }^{1\mathrm{/}8-ϵ}$ for any $ϵ>0$.