In this thesis we study quantum variance for the modular surface X. This is an important problem in mathematical physics and number theory concerning the mass equidistribution of Maass-Hecke cusp forms on the arithmetic hyperbolic surfaces. We evaluate asymptotically the quantum variance, which is introduced by S. Zelditch and describes the fluctuations of a quantum observable. We show that the quantum variance is equal to the classical variance of the geodesic flow on the unit cotangent bundle of X, but twisted by the central value of the Maass-Hecke L-functions.
Our approach is via Poincare series and Kuznetsov trace formula, which transfer the spectral sum into the sum of Kloosterman sums. The treatment of the non-diagonal terms contributions is subtle and forms the core of this thesis. It turns out that the continuous spectrum part is not negligible and contributes to the main term, but in general, it is small in the cuspidal subspace. We then make use of Watson's explicit triple product formula to determine the leading term in the asymptotic formula for the quantum variance and analyze its structure, which leads to our main result.