We will show that the relation of the heat kernels for the Schro dinger operators with uniform magnetic fields on the hyperbolic plane H 2 (the Maass Laplacians) and for the Schro dinger operators with Morse potentials on R is given by means of a one-dimensional Fourier transform in the framework of stochastic analysis, where the Brownian motion on H 2 plays 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.
Academic Press
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.