Laplace-Beltrami operator on a Riemann surface and equidistribution of measures

We consider the Laplace-Beltrami operator on a compact Riemann surface of a constant negative curvature. For any eigenvalue of the Laplace-Beltrami operator there is an associated sequence of measures on the Riemann surface. These measures naturally appear in Quantum Chaos type questions in the theory of electro-magnetic flow on a Riemann surface. The main result of the paper is the claim that this sequence of measures has the Liouville measure as the (weak*) limit. We prove a quantitative version of this equidistribution claim.