arXiv subjects
Publications and source records attributed to Frederic Faure.
We show that for smooth contact Anosov flows in dimension 3, the resonant states associated to the first band of Ruelle resonances are distributions that are killed by the unstable derivative.
In this paper, we show that some spectral properties of Anosov diffeomorphisms can be obtained by semi-classical analysis. In particular the Ruelle resonances which are eigenvalues of the Ruelle transfer operator acting in suitable anisotropic Sobolev spaces and which govern the decay of dynamical correlations, can be treated as the quantum resonances of open quantum systems in the Aguilar-Baslev-Combes theory or the more recent Helffer-Sjostrand phase-space theory.
Prequantum dynamics has been introduced in the 70' by Kostant-Souriau-Kirillov as an intermediate between classical and quantum dynamics. In common with the classical dynamics, prequantum dynamics transports functions on phase space, but add some phases which are important in quantum interference effects. In the case of hyperbolic dynamical systems, it is believed that the study of the prequantum dynamics will give a better understanding of the quantum interference effects for large time, and of their statistical properties. We consider a linear hyperbolic map M in SL(2,Z) which generates a chaotic dynamics on the torus. This dynamics is lifted on a prequantum fiber bundle. This gives a unitary prequantum (partially hyperbolic) map. We calculate its resonances and show that they are related with the quantum eigenvalues. A remarkable consequence is that quantum dynamics emerges from long time behaviour of prequantum dynamics. We present trace formulas, and discuss perspectives of this approach in the non linear case.
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 Mq. The usual semi-classical Trace formula expresses Tr(Mq^t) for finite time t, in the limit hbar->0, in terms of periodic orbits of M of period t. Recent work reach time t<< tE/6 where tE=log(1/hbar)/lambda is the Ehrenfest time, and lambda is the Lyapounov coefficient. Using a semi-classical normal form description of the dynamics uniformly over phase space, we show how to extend the trace formula for longer time of the form t= C.tE where C is any constant, with an arbitrary small error.
We study two simple real analytic uniformly hyperbolic dynamical systems: expanding maps on the circle S1 and hyperbolic maps on the torus T2. 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 L2(T2) respectively. The trace class operator is obtained by conjugation of the Ruelle transfer operator in a similar way quantum resonances are obtained in open quantum systems. We comment this analogy.
We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of ``strong scarring''), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound 1/2.
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 bundles topology. The formation of topologically coupled bands is demonstrated. This phenomenon is expected to be present in many quantum systems.
Topological Chern indices are related to the number of rotational states in each molecular vibrational band. Modification of the indices is associated to the appearance of ``band degeneracies'', and exchange of rotational states between two consecutive bands. The topological dynamical origin of these indices is proven through a semi-classical approach, and their values are computed in two examples. The relation with the integer quantum Hall effect is briefly discussed.
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 semi-classical approach with 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 (θ) 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.
The semi-classical study of the integer Quantum Hall conductivity is investigated for electrons in a bi-periodic potential $V(x,y)$. The Hall conductivity is due to the tunnelling effect and we concentrate our study to potentials having three wells in a periodic cell. A non-zero topological conductivity requires special conditions for the positions, and shapes of the wells. The results are derived analytically and well confirmed by numerical calculations.