11 research outputs found
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
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
Resolvent estimates for normally hyperbolic trapped sets
We give pole free strips and estimates for resolvents of semiclassical
operators which, on the level of the classical flow, have normally hyperbolic
smooth trapped sets of codimension two in phase space. Such trapped sets are
structurally stable and our motivation comes partly from considering the wave
equation for Kerr black holes and their perturbations, whose trapped sets have
precisely this structure. We give applications including local smoothing
effects with epsilon derivative loss for the Schr\"odinger propagator as well
as local energy decay results for the wave equation.Comment: Further changes to erratum correcting small problems with Section 3.5
and Lemma 4.1; this now also corrects hypotheses, explicitly requiring
trapped set to be symplectic. Erratum follows references in this versio
Fractal Weyl law for Linux Kernel Architecture
We study the properties of spectrum and eigenstates of the Google matrix of a
directed network formed by the procedure calls in the Linux Kernel. Our results
obtained for various versions of the Linux Kernel show that the spectrum is
characterized by the fractal Weyl law established recently for systems of
quantum chaotic scattering and the Perron-Frobenius operators of dynamical
maps. The fractal Weyl exponent is found to be that
corresponds to the fractal dimension of the network . The
eigenmodes of the Google matrix of Linux Kernel are localized on certain
principal nodes. We argue that the fractal Weyl law should be generic for
directed networks with the fractal dimension .Comment: RevTex 6 pages, 7 figs, linked to arXiv:1003.5455[cs.SE]. Research at
http://www.quantware.ups-tlse.fr/, Improved version, changed forma
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
Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum
A quantum version of transition state theory based on a quantum normal form
(QNF) expansion about a saddle-centre-...-centre equilibrium point is
presented. A general algorithm is provided which allows one to explictly
compute QNF to any desired order. This leads to an efficient procedure to
compute quantum reaction rates and the associated Gamov-Siegert resonances. In
the classical limit the QNF reduces to the classical normal form which leads to
the recently developed phase space realisation of Wigner's transition state
theory. It is shown that the phase space structures that govern the classical
reaction d ynamicsform a skeleton for the quantum scattering and resonance
wavefunctions which can also be computed from the QNF. Several examples are
worked out explicitly to illustrate the efficiency of the procedure presented.Comment: 132 pages, 31 figures, corrected version, Nonlinearity, 21 (2008)
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