Semiclassical structure of chaotic resonance eigenfunctions

We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the non-unitary quantum propagator, and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as $\hbar\to 0$ on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates $\{\psi(\hbar)\}_{\hbar\to 0}$ is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker map, for which the probability density in position space is observed to have self-similarity properties.Comment: 4 pages, 4 figures; some minor corrections, some changes in presentatio

Quantum electrodynamics of relativistic bound states with cutoffs

We consider an Hamiltonian with ultraviolet and infrared cutoffs, describing the interaction of relativistic electrons and positrons in the Coulomb potential with photons in Coulomb gauge. The interaction includes both interaction of the current density with transversal photons and the Coulomb interaction of charge density with itself. We prove that the Hamiltonian is self-adjoint and has a ground state for sufficiently small coupling constants.Comment: To appear in "Journal of Hyperbolic Differential Equation

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

Quantum ergodicity for Pauli Hamiltonians with spin 1/2

Quantum ergodicity, which expresses the semiclassical convergence of almost all expectation values of observables in eigenstates of the quantum Hamiltonian to the corresponding classical microcanonical average, is proven for non-relativistic quantum particles with spin 1/2. It is shown that quantum ergodicity holds, if a suitable combination of the classical translational dynamics and the spin dynamics along the trajectories of the translational motion is ergodic.Comment: 20 pages, no figure

Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-, II

We do the spectral analysis of the Hamiltonian for the weak leptonic decay of the gauge bosons W+/-. Using Mourre theory, it is shown that the spectrum between the unique ground state and the first threshold is purely absolutely continuous. Neither sharp neutrino high energy cutoff nor infrared regularization are assumed.Comment: To appear in Ann. Henri Poincar\'

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

Entropy of semiclassical measures for nonpositively curved surfaces

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. We follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus on the main differences and refer the reader to (arXiv:0809.0230) for the details of analogous lemmas.Comment: 20 pages. This note provides a detailed proof of a result announced in appendix A of a previous work (arXiv:0809.0230, version 2

Probabilistic Weyl laws for quantized tori

For the Toeplitz quantization of complex-valued functions on a $2n$-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

Hamiltonian dynamics and spectral theory for spin-oscillators

We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have infinitely many transversally elliptic singularities, exactly one elliptic-elliptic singularity and one focus-focus singularity. The most interesting dynamical features of integrable systems, and in particular of spin-oscillators, are encoded in their singularities. In the first part of the paper we study the symplectic dynamics around the focus-focus singularity. In the second part of the paper we quantize the coupled spin-oscillators systems and study their spectral theory. The paper combines techniques from semiclassical analysis with differential geometric methods.Comment: 32 page

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