Fractal Weyl laws for chaotic open systems

We present a result relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. The result is supported by numerical computation of the resonances of the system of n disks on a plane. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.Comment: revtex4, 4 pages, 3 figure

Numerical Study of Quantum Resonances in Chaotic Scattering

This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy $E$ scales like $\hbar^{-\frac{D(K_E)+1}{2}}$ as $\hbar\to{0}$. Here, $K_E$ denotes the subset of the classical energy surface $\{H=E\}$ which stays bounded for all time under the flow generated by the Hamiltonian $H$ and $D(K_E)$ denotes its fractal dimension. Since the number of bound states in a quantum system with $n$ degrees of freedom scales like $\hbar^{-n}$, this suggests that the quantity $\frac{D(K_E)+1}{2}$ represents the effective number of degrees of freedom in scattering problems.Comment: 24 pages, including 44 figure

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the one-dimensional stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov--Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.Comment: 18 pages, no figure

Limiting Carleman weights and anisotropic inverse problems

In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic X-ray transform. Earlier results in dimension $n \geq 3$ were restricted to real-analytic metrics.Comment: 58 page

Distribution of resonances for open quantum maps

We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.Comment: 60 pages, no figures (numerical results are shown in other references

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

Asymptotic Density of Eigenvalue Clusters for the Perturbed Landau Hamiltonian

We consider the Landau Hamiltonian (i.e. the 2D Schroedinger operator with constant magnetic field) perturbed by an electric potential V which decays sufficiently fast at infinity. The spectrum of the perturbed Hamiltonian consists of clusters of eigenvalues which accumulate to the Landau levels. Applying a suitable version of the anti-Wick quantization, we investigate the asymptotic distribution of the eigenvalues within a given cluster as the number of the cluster tends to infinity. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the Radon transform of the perturbation potential V.Comment: 30 pages. The explicit dependence on B and V in Theorem 1.6 (i) - (ii) indicated. Typos corrected. To appear in Communications in Mathematical Physic

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) R1-R11

Semi-classical analysis and passive imaging

Passive imaging is a new technique which has been proved to be very efficient, for example in seismology: the correlation of the noisy fields, computed from the fields recorded at different points, is strongly related to the Green function of the wave propagation. The aim of this paper is to provide a mathematical context for this approach and to show, in particular, how the methods of semi-classical analysis can be be used in order to find the asymptotic behaviour of the correlations.Comment: Invited paper to appear in NONLINEARITY; Accepted Revised versio

War and first onset of suicidality: the role of mental disorders

Background Suicide rates increase following periods of war; however, the mechanism through which this occurs is not known. The aim of this paper is to shed some light on the associations of war exposure, mental disorders, and subsequent suicidal behavior. Method A national sample of Lebanese adults was administered the Composite International Diagnostic Interview to collect data on lifetime prevalence and age of onset of suicide ideation, plan, and attempt, and mental disorders, in addition to information about exposure to stressors associated with the 1975–1989 Lebanon war. Results The onset of suicide ideation, plan, and attempt was associated with female gender, younger age, post-war period, major depression, impulse-control disorders, and social phobia. The effect of post-war period on each type of suicide outcome was largely explained by the post-war onset of mental disorders. Finally, the conjunction of having a prior impulse-control disorder and either being a civilian in a terror region or witnessing war-related stressors was associated with especially high risk of suicide attempt. Conclusions The association of war with increased risk of suicidality appears to be partially explained by the emergence of mental disorders in the context of war. Exposure to war may exacerbate disinhibition among those who have prior impulse-control disorders, thus magnifying the association of mental disorders with suicidality.Psycholog