Brownian Motion on the Hyperbolic Plane and Selberg Trace Formula

We will show that the relation of the heat kernels for the Schrِdinger operators with uniform magnetic fields on the hyperbolic planeH2(the Maass Laplacians) and for the Schrِdinger operators with Morse potentials onRis given by means of a one-dimensional Fourier transform in the framework of stochastic analysis, where the Brownian motion onH2plays an important role. By using this relation, we will give the explicit forms of the Green functions. As a typical related problem, we will discuss the Selberg trace formula. The close relation of the trace formula with the corresponding classical mechanics will also be discussed.