8,429 research outputs found

    Wannier-Stark resonances in optical and semiconductor superlattices

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    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

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    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 XX 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

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    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

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    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 XX which is acted on by any continuous semigroup {S(t)}t≥0\{S(t)\}_{t \geq 0}. Suppose that §(t)}t≥0\S(t)\}_{t \geq 0} possesses a global attractor A\mathcal{A}. We show that, for any generalized Banach limit LIMT→∞\underset{T \rightarrow \infty}{\rm{LIM}} and any distribution of initial conditions m0\mathfrak{m}_0, that there exists an invariant probability measure m\mathfrak{m}, whose support is contained in A\mathcal{A}, such that ∫Xϕ(x)dm(x)=LIMT→∞1T∫0T∫Xϕ(S(t)x)dm0(x)dt, \int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t, for all observables ϕ\phi living in a suitable function space of continuous mappings on XX. 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 {S(t)}t≥0\{S(t)\}_{t \geq 0} 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 XX 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

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    Using a weighted HsH^s-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 LL 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 LL 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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