Frédéric Faure

Upper Bound on the Density of Ruelle Resonances for Anosov Flows

Using a semiclassical approach we show that the spectrum of a smooth Anosov vector field V on a compact manifold is discrete (in suitable anisotropic Sobolev spaces) and then we provide an upper bound for the density of eigenvalues of the operator (−i)V, called Ruelle resonances, close to the real axis and for large real parts.RésuméPar une approche semiclassique on montre que le spectre d’un champ de vecteur d’Anosov V sur une variété compacte est discret (dans des espaces de Sobolev anisotropes adaptés). On montre ensuite une majoration de la densité de valeurs propres de l’opérateur (−i)V, appelées résonances de Ruelle, près de l’axe réel et pour les grandes parties réelles.

Ruelle?Pollicott resonances for real analytic hyperbolic maps

We study two simple real analytic uniformly hyperbolic dynamical systems: expanding maps on the circle S1 and hyperbolic maps on the torus . We show that the Ruelle–Pollicott resonances which describe time correlation functions of the chaotic dynamics can be obtained as the eigenvalues of a trace class operator in Hilbert space L2(S1) or , respectively. The trace class operator is obtained by conjugation of the Ruelle transfer operator in a similar way to how quantum resonances are obtained in open quantum systems. We comment on this analogy.

Topologically coupled energy bands in molecules

Abstract We propose a concrete application of the Atiyah–Singer index formula in molecular physics, giving the exact number of levels in energy bands, in terms of vector bundle topology. The formation of topologically coupled bands is demonstrated. This phenomenon is expected to be present in many quantum systems.

Power spectrum of the geodesic flow on hyperbolic manifolds

We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.

Fractional monodromy in systems with coupled angular momenta

We present a one-parameter family of systems with fractional monodromy, which arises from a 1:2 diagonal action of a dynamical symmetry SO(2) .I n a regime of adiabatic separation of slow and fast motions, we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi-quantum spectra.We present a 1-parameter family of systems with fractional monodromy and adiabatic separation of motion. We relate the presence of monodromy to a redistribution of states both in the quantum and semi-quantum spectrum. We show how the fractional monodromy arises from the non diagonal action of the dynamical symmetry of the system and manifests itself as a generic property of an important subclass of adiabatically coupled systems.

Topological properties of quantum periodic Hamiltonians

We consider periodic quantum Hamiltonians on the torus phase space (Harper-like Hamiltonians). We calculate the topological Chern index which characterizes each spectral band in the generic case. This calculation is made by a semiclassical approach with the use of quasi- modes. As a result, the Chern index is equal to the homotopy of the path of these quasi-modes on phase space as the Floquet parameter of the band is varied. It is quite interesting that the Chern indices, defined as topological quantum numbers, can be expressed from simple properties of the classical trajectories.

Qualitative Features of Intra-molecular Dynamics. What can be Learned from Symmetry and Topology?

Mathematical tools which are appropriate for the qualitative analysis of simple molecular quantum systems are briefly introduced and discussed.

Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

We consider a simple model of an open partially expanding map. Its trapped set

Loss of quantum coherence in a system coupled to a zero-temperature environment

{\mathcal{K}}

Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum

in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component