8,429 research outputs found
Wannier-Stark resonances in optical and semiconductor superlattices
In this work, we discuss the resonance states of a quantum particle in a
periodic potential plus a static force. Originally this problem was formulated
for a crystal electron subject to a static electric field and it is nowadays
known as the Wannier-Stark problem. We describe a novel approach to the
Wannier-Stark problem developed in recent years. This approach allows to
compute the complex energy spectrum of a Wannier-Stark system as the poles of a
rigorously constructed scattering matrix and solves the Wannier-Stark problem
without any approximation. The suggested method is very efficient from the
numerical point of view and has proven to be a powerful analytic tool for
Wannier-Stark resonances appearing in different physical systems such as
optical lattices or semiconductor superlattices.Comment: 94 pages, 41 figures, typos corrected, references adde
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
Anomalous diffusion in the resonant quantum kicked rotor
We study the resonances of the quantum kicked rotor subjected to an
excitation that follows a deterministic time-dependent prescription. For the
primary resonances we find an analytical relation between the long-time
behavior of the standard deviation and the external kick strength. For the
secondary resonances we obtain essentially the same result numerically.
Selecting the time sequence of the kick allows to obtain a variety of
asymptotic wave-function spreadings: super-ballistic, ballistic, sub-ballistic,
diffusive, sub-diffusive and localized.Comment: 5 pages, 3 figures To appear in Physica A
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
Existence and sharp localization in velocity of small-amplitude Boltzmann shocks
Using a weighted -contraction mapping argument based on the macro-micro
decomposition of Liu and Yu, we give an elementary proof of existence, with
sharp rates of decay and distance from the Chapman--Enskog approximation, of
small-amplitude shock profiles of the Boltzmann equation with hard-sphere
potential, recovering and slightly sharpening results obtained by Caflisch and
Nicolaenko using different techniques. A key technical point in both analyses
is that the linearized collision operator is negative definite on its
range, not only in the standard square-root Maxwellian weighted norm for which
it is self-adjoint, but also in norms with nearby weights. Exploring this issue
further, we show that is negative definite on its range in a much wider
class of norms including norms with weights asymptotic nearly to a full
Maxwellian rather than its square root. This yields sharp localization in
velocity at near-Maxwellian rate, rather than the square-root rate obtained in
previous analyse
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