48 research outputs found
Entropy of chaotic eigenstates
These notes present a recent approach to study the high-frequency eigenstates
of the Laplacian on compact Riemannian manifolds of negative sectional
curvature. The main result is a lower bound on the Kolmogorov-Sinai entropy of
the semiclassical measures associated with sequences of eigenstates, showing
that high-frequency eigenstates cannot be too localized. The method is extended
to the case of semiclassical Hamiltonian operators for which the classical flow
in some energy range is of Anosov type, and to the case of quantized Anosov
diffeomorphisms on the torus.Comment: Notes of the minicourse given at the workshop "Spectrum and
dynamics", Centre de Recherches Mathematiques, Montreal, April 2008
Quantized open chaotic systems
Two different "wave chaotic" systems, involving complex eigenvalues or
resonances, can be analyzed using common semiclassical methods. In particular,
one obtains fractal Weyl upper bounds for the density of resonances/eigenvalues
near the real axis, and a classical dynamical criterion for a spectral gap.Comment: Proceedings of the conference QMath 11; QMath 11, Hradec Kralove :
Czech Republic (2010
Quantum transfer operators and chaotic scattering
Transfer operators have been used widely to study the long time properties of
chaotic maps or flows. We describe quantum analogues of these operators, which
have been used as toy models by the quantum chaos community, but are also
relevant to study "physical" continuous time systems
Spectral theory of damped quantum chaotic systems
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 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 defined quantities, like the value
distribution of the time-averaged damping. We also present a new condition for
a spectral gap, depending on the set of minimally damped trajectories.Comment: Lecture given at the Journ\'ees semiclassiques 2011, Biarritz, 6-10
June 201
Quantum transfer operators and quantum scattering
These notes describe a new method to investigate the spectral properties if
quantum scattering Hamiltonians, developed in collaboration with J. Sj\"ostrand
and M.Zworski. This method consists in constructing a family of "quantized
transfer operators" associated with a classical Poincar\'e section
near some fixed classical energy E. These operators are finite dimensional, and
have the structure of "open quantum maps". In the semiclassical limit, the
family encode the quantum dynamics near the energy E. In
particular, the quantum resonances of the form , for , are
obtained as the roots of .Comment: 18 pages, 3 figure
Fractal Weyl laws in discrete models of chaotic scattering
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 r\'{e}gime. 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 the (physically more
relevant) case of Hamiltonian chaotic scattering. Within a simplified model, we
are able to rigorously prove this Weyl law, and compute quantities related to
the "coherent transport" through the system, namely the conductance and "shot
noise". The latter is close to the prediction of random matrix theory.Comment: Invited article in the Special Issue of Journal of Physics A on
"Trends in Quantum Chaotic Scattering
Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
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.Comment: We added the proof of the Entropic Uncertainty Principle. 45 pages, 2
EPS figure
Chaotic vibrations and strong scars
This article aims at popularizing some aspects of "quantum chaos", in
particular the study of eigenmodes of classically chaotic systems, in the
semiclassical (or high frequency) limit
Eigenmodes of the damped wave equation and small hyperbolic subsets
We study stationary solutions of the damped wave equation on a compact and
smooth Riemannian manifold without boundary. In the high frequency limit, we
prove that a sequence of -damped stationary solutions cannot be
completely concentrated in small neighborhoods of a small fixed hyperbolic
subset made of -damped trajectories of the geodesic flow. The article
also includes an appendix (by S. Nonnenmacher and the author) where we
establish the existence of an inverse logarithmic strip without eigenvalues
below the real axis, under a pressure condition on the set of undamped
trajectories.Comment: 24 pages. With an appendix by S. Nonnenmacher and the author. In this
new version, we modified an uncorrect exponent in the statement of Theorem
A.1 from the appendi