Selberg's Trace Formula: An Introduction

Abstract

The aim of this short lecture course is to develop Selberg's trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the Laplace-Beltrami operator –Δ on the surface M represents the quantum Hamiltonian of a particle, whose classical dynamics is governed by the (strongly chaotic) geodesic flow on the unit tangent bundle of M. The trace formula is currently the only available tool to analyze the fine structure of the spectrum of –Δ; no individual formulas for its eigenvalues are known. In the case of more general quantum systems, the role of Selberg's formula is taken over by the semiclassical Gutzwiller trace formula [11], [7].

Locations

• arXiv (Cornell University) - View - PDF
• Cambridge University Press eBooks - View