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

Yves Colin de Verdière

Displaying similar documents to “Spectrum of the Laplace operator and periodic geodesics: thirty years after”

Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Nalini Anantharaman, Stéphane Nonnenmacher (2007)

Annales de l’institut Fourier

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We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized. ...

Semi-classical formula beyond the Ehrenfest time in quantum chaos. (I) Trace formula

Frédéric Faure (2007)

Annales de l’institut Fourier

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We consider a nonlinear area preserving Anosov map $M$ on the torus phase space, which is the simplest example of a fully chaotic dynamics. We are interested in the quantum dynamics for long time, generated by the unitary quantum propagator $\hat{M}$ . The usual semi-classical Trace formula expresses $T r ({\hat{M}}^{t})$ for finite time $t$ , in the limit $ℏ \to 0$ , in terms of periodic orbits of $M$ of period $t$ . Recent work reach time $t ≪ t_{E} / 6$ where $t_{E} = log (1 / ℏ) / λ$ is the Ehrenfest time, and $λ$ is the Lyapounov coefficient. Using a semi-classical...

Elementary linear algebra for advanced spectral problems

Johannes Sjöstrand, Maciej Zworski (2007)

Annales de l’institut Fourier

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We describe a simple linear algebra idea which has been used in different branches of mathematics such as bifurcation theory, partial differential equations and numerical analysis. Under the name of the Schur complement method it is one of the standard tools of applied linear algebra. In PDE and spectral analysis it is sometimes called the Grushin problem method, and here we concentrate on its uses in the study of infinite dimensional problems, coming from partial differential operators...

Generic Nekhoroshev theory without small divisors

Abed Bounemoura, Laurent Niederman (2012)

Annales de l’institut Fourier

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In this article, we present a new approach of Nekhoroshev’s theory for a generic unperturbed Hamiltonian which completely avoids small divisors problems. The proof is an extension of a method introduced by P. Lochak, it combines averaging along periodic orbits with simultaneous Diophantine approximation and uses geometric arguments designed by the second author to handle generic integrable Hamiltonians. This method allows to deal with generic non-analytic Hamiltonians and to obtain new...

Formal geometric quantization

Paul-Émile Paradan (2009)

Annales de l’institut Fourier

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Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $(M, Ω)$ which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map $Φ$ is proper so that the reduced space $M_{μ} : = Φ^{- 1} (K \cdot μ) / K$ is compact for all $μ$ . Then, we can define the “formal geometric quantization” of $M$ as $𝒬_{K}^{- \infty} (M) : = \sum_{μ \in \hat{K}} 𝒬 (M_{μ}) V_{μ}^{K} .$ The aim of this article is to study the functorial properties of the assignment $(M, K) \to 𝒬_{K}^{- \infty} (M)$ .