Spectrum of the laplace operator and periodic geodesics: Thirty years after

Abstract

What is called the "Semi-classical trace formula" is a formula expressing the smoothed density of states of the Laplace operator on a compact Riemannian manifold in terms of the periodic geodesics. Mathematical derivation of such formulas were provided in the seventies by several authors. The main goal of this paper is to state the formula and to give a self-contained proof independent of the difficult use of the global calculus of Fourier Integral Operators. This proof is close in the spirit of the first proof given in the authors thesis. It uses the time-dependent Schrödinger equation, some facts about the geodesic flow, the stationary phase approximation and the metaplectic representation as a computational tool.