Subconvexity Implies Effective Quantum Unique Ergodicity for Hecke-Maa{\ss} Cusp Forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathrm{SL}_2(\mathbb{R})$

Abstract

It is a folklore result in arithmetic quantum chaos that quantum unique ergodicity on the modular surface with an effective rate of convergence follows from subconvex bounds for certain triple product $L$-functions. The physical space manifestation of this result, namely the equidistribution of mass of Hecke-Maass cusp forms, was proven to follow from subconvexity by Watson, whereas the phase space manifestation of quantum unique ergodicity has only previously appeared in the literature for Eisenstein series via work of Jakobson. We detail the analogous phase space result for Hecke-Maass cusp forms. The proof relies on the Watson-Ichino triple product formula together with a careful analysis of certain archimedean integrals of Whittaker functions.