Quantum Variance of Maass-Hecke Cusp Forms

Abstract

In this paper we study quantum variance for the modular surface \({X=\Gamma\backslash\mathbb{H}}\), where \({\Gamma=SL_2(\mathbb{Z})}\) is the full modular group. We evaluate asymptotically the quantum variance, which is introduced by S. Zelditch and describes the fluctuations of a quantum observable. It is shown that the quantum variance is equal to the classical variance of the geodesic flow on S*X, the unit cotangent bundle of X, but twisted by the central value of the Maass-Hecke L-functions.