Spectrum of the Laplace operator and periodic geodesics: thirty years after

Colin de Verdière, Yves

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2429-2463.

On appelle « Formule de trace semi-classique » une formule exprimant la densité d’état régularisée du laplacien d’une variété riemannienne compacte en termes de ses géodésiques périodiques. Des preuves de telles formules ont été données par plusieurs auteurs dans les années 70. Le but principal de cet article est de présenter cette formule et d’en donner une preuve complète et indépendante du difficile calcul global des opérateurs intégraux de Fourier. Cette preuve est d’un esprit assez proche de celle de la thèse de l’auteur. Elle utilise l’équation de Schrödinger dépendant du temps, des propriétés des géodésiques, la méthode de la phase stationnaire et la représentation métaplectique comme outil de calcul.

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.

Classification : 35P20, 53C22, 58J40
Keywords: Laplace operator, semi-classics, symplectic geometry, twist map, trace formula, spectrum, periodic geodesics, metaplectic, determinant
Mot clés : Laplacien, semi-classique, géométrie symplectique, application twist, formule de trace, spectre, géodésiques periodiques, métaplectique, déterminant

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