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

Nalini Anantharaman; Stéphane Nonnenmacher

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Spectral theory of damped quantum chaotic systems

Stéphane Nonnenmacher (2011)

Journées Équations aux dérivées partielles

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We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on $X$ and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically...

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

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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...

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Propagation through trapped sets and semiclassical resolvent estimates

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Annales de l’institut Fourier

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Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.

Quadratic uniformity of the Möbius function

Ben Green, Terence Tao (2008)

Annales de l’institut Fourier

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