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(1.1) P (h) = −h∆+ V (x) , V ∈ C c (X) , X = R 2 , with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given… (More)

We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows [23],[46],[47], classical… (More)

In this paper we construct a sequence of eigenfunctions of the “quantum Arnold’s cat map” that, in the semiclassical limit, show a strong scarring phenomenon on the periodic orbits of the dynamics. More precisely, those states have a semiclassical limit measure that is the sum of 1/2 the normalized Lebesgue measure on the torus plus 1/2 the normalized Dirac… (More)

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… (More)

Using the Bargmann–Husimi representation of quantum mechanics on a torus phase space, we study analytically eigenstates of quantized cat maps [9]. The linearity of these maps implies a close relationship between classically invariant sublattices on the one hand, and the patterns (or ‘constellations’) of Husimi zeros of certain quantum eigenstates on the… (More)

We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states, as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of… (More)

1.1. Statement of the results. In this note we analyze simple models of classical chaotic open systems and of their quantizations. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller of the system. In a simplified… (More)

We analyze simple models of quantum chaotic scattering, namely quantized open baker’s maps. We numerically compute the density of quantum resonances in the semiclassical régime. This density satisfies a fractal Weyl law, where the exponent is governed by the (fractal) dimension of the set of trapped trajectories. This type of behaviour is also expected in… (More)

These notes present a description of quantum chaotic eigenstates, that is bound states of quantum dynamical systems, whose classical limit is chaotic. The classical dynamical systems we will be dealing with are mostly of two types: geodesic flows on Euclidean domains (“billiards”) or compact riemannian manifolds, and canonical transformations on a compact… (More)

- Albert Fannjiang, Stéphane Nonnenmacher, Lech Wołowski
- 2004

We consider the effect of noise on the dynamics generated by volumepreserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of… (More)