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

Yves Colin de Verdière[1]

  • [1] Institut Fourier Unité mixte de recherche CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères Cedex (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 7, page 2429-2463
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Colin de Verdière, Yves. "Spectrum of the Laplace operator and periodic geodesics: thirty years after." Annales de l’institut Fourier 57.7 (2007): 2429-2463. <http://eudml.org/doc/10303>.

@article{ColindeVerdière2007,
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.},
affiliation = {Institut Fourier Unité mixte de recherche CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères Cedex (France)},
author = {Colin de Verdière, Yves},
journal = {Annales de l’institut Fourier},
keywords = {Laplace operator; semi-classics; symplectic geometry; twist map; trace formula; spectrum; periodic geodesics; metaplectic; determinant; semiclassical trace formulas},
language = {eng},
number = {7},
pages = {2429-2463},
publisher = {Association des Annales de l’institut Fourier},
title = {Spectrum of the Laplace operator and periodic geodesics: thirty years after},
url = {http://eudml.org/doc/10303},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Colin de Verdière, Yves
TI - Spectrum of the Laplace operator and periodic geodesics: thirty years after
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2429
EP - 2463
AB - 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.
LA - eng
KW - Laplace operator; semi-classics; symplectic geometry; twist map; trace formula; spectrum; periodic geodesics; metaplectic; determinant; semiclassical trace formulas
UR - http://eudml.org/doc/10303
ER -

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