✦ LIBER ✦

Brownian Motion on the Hyperbolic Plane and Selberg Trace Formula

✍ Scribed by Nobuyuki Ikeda; Hiroyuki Matsumoto

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 301 KB
Volume: 163
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1998.3382

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✦ Synopsis

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.

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## Abstract Consider the Poincare unit disk model for the hyperbolic plane **H**^2^. Let Ξ be the set of all horocycles in **H**^2^ parametrized by (__θ, p__), where __e^iθ^__ is the point where a horocycle __ξ__ is tangent to the boundary |__z__| = 1, and __p__ is the hyperbolic distance from __ξ_