377 research outputs found
On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
Given any fixed positive semi-definite diagonal matrix
we derive the explicit formula for the density of complex eigenvalues for
random matrices of the form } where the random unitary
matrices are distributed on the group according to the Haar
measure.Comment: 10 pages, 1 figur
Coarse Grained Liouville Dynamics of piecewise linear discontinuous maps
We compute the spectrum of the classical and quantum mechanical
coarse-grained propagators for a piecewise linear discontinuous map. We analyze
the quantum - classical correspondence and the evolution of the spectrum with
increasing resolution. Our results are compared to the ones obtained for a
mixed system.Comment: 11 pages, 8 figure
On the resonance eigenstates of an open quantum baker map
We study the resonance eigenstates of a particular quantization of the open
baker map. For any admissible value of Planck's constant, the corresponding
quantum map is a subunitary matrix, and the nonzero component of its spectrum
is contained inside an annulus in the complex plane, . We consider semiclassical sequences of eigenstates, such that the
moduli of their eigenvalues converge to a fixed radius . We prove that, if
the moduli converge to , then the sequence of eigenstates
converges to a fixed phase space measure . The same holds for
sequences with eigenvalue moduli converging to , with a different
limit measure . Both these limiting measures are supported on
fractal sets, which are trapped sets of the classical dynamics. For a general
radius , we identify families of eigenstates with
precise self-similar properties.Comment: 32 pages, 2 figure
Resonance distribution in open quantum chaotic systems
In order to study the resonance spectra of chaotic cavities subject to some
damping (which can be due to absorption or partial reflection at the
boundaries), we use a model of damped quantum maps. In the high-frequency
limit, the distribution of (quantum) decay rates is shown to cluster near a
``typical'' value, which is larger than the classical decay rate of the
corresponding damped ray dynamics. The speed of this clustering may be quite
slow, which could explain why it has not been detected in previous numerical
data.Comment: 4 pages. Compared with version 2, we have slightly modified the
figures, corrected some misprints, and added the values for the fits in
figure
Some open questions in "wave chaos"
The subject area referred to as "wave chaos", "quantum chaos" or "quantum
chaology" has been investigated mostly by the theoretical physics community in
the last 30 years. The questions it raises have more recently also attracted
the attention of mathematicians and mathematical physicists, due to connections
with number theory, graph theory, Riemannian, hyperbolic or complex geometry,
classical dynamical systems, probability etc. After giving a rough account on
"what is quantum chaos?", I intend to list some pending questions, some of them
having been raised a long time ago, some others more recent
Probabilistic Weyl laws for quantized tori
For the Toeplitz quantization of complex-valued functions on a
-dimensional torus we prove that the expected number of eigenvalues of
small random perturbations of a quantized observable satisfies a natural Weyl
law. In numerical experiments the same Weyl law also holds for ``false''
eigenvalues created by pseudospectral effects.Comment: 33 pages, 3 figures, v2 corrected listed titl
Fractal Weyl law for quantum fractal eigenstates
The properties of the resonant Gamow states are studied numerically in the
semiclassical limit for the quantum Chirikov standard map with absorption. It
is shown that the number of such states is described by the fractal Weyl law
and their Husimi distributions closely follow the strange repeller set formed
by classical orbits nonescaping in future times. For large matrices the
distribution of escape rates converges to a fixed shape profile characterized
by a spectral gap related to the classical escape rate.Comment: 4 pages, 5 figs, minor modifications, research at
http://www.quantware.ups-tlse.fr
Spectral problems in open quantum chaos
This review article will present some recent results and methods in the study
of 1-particle quantum or wave scattering systems, in the semiclassical/high
frequency limit, in cases where the corresponding classical/ray dynamics is
chaotic. We will focus on the distribution of quantum resonances, and the
structure of the corresponding metastable states. Our study includes the toy
model of open quantum maps, as well as the recent quantum monodromy operator
method.Comment: Compared with the previous version, misprints and typos have been
corrected, and the bibliography update
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.Comment: 57 page
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving 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 various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
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