117 research outputs found
Spectral shift function for operators with crossed magnetic and electric fields
We obtain a representation formula for the derivative of the spectral shift
function related to the operators and . We establish a limiting absorption principle
for and an estimate for
, provided , where $Q =
(D_x - By)^2 + D_y^2 + V(x,y).
Classical limit of the quantum Zeno effect
The evolution of a quantum system subjected to infinitely many measurements
in a finite time interval is confined in a proper subspace of the Hilbert
space. This phenomenon is called "quantum Zeno effect": a particle under
intensive observation does not evolve. This effect is at variance with the
classical evolution, which obviously is not affected by any observations. By a
semiclassical analysis we will show that the quantum Zeno effect vanishes at
all orders, when the Planck constant tends to zero, and thus it is a purely
quantum phenomenon without classical analog, at the same level of tunneling.Comment: 10 pages, 2 figure
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
on the forward (backward) trapped set of the classical dynamics.
The limit of a sequence of eigenstates 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
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 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
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
Spectral Analysis of a Two Body Problem with Zero Range Perturbation
We consider a class of singular, zero-range perturbations of the Hamiltonian
of a quantum system composed by a test particle and a harmonic oscillators in
dimension one, two and three and we study its spectrum. In facts we give a
detailed characterization of point spectrum and its asymptotic behavior with
respect to the parameters entering the Hamiltonian. We also partially describe
the positive spectrum and scattering properties of the Hamiltonian.Comment: Version submitted for publication, AMStex, 22 page
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
Semiclassical measures and the Schroedinger flow on Riemannian manifolds
In this article we study limits of Wigner distributions (the so-called
semiclassical measures) corresponding to sequences of solutions to the
semiclassical Schroedinger equation at times scales tending to
infinity as the semiclassical parameter tends to zero (when this is equivalent to consider solutions to the non-semiclassical
Schreodinger equation). Some general results are presented, among which a weak
version of Egorov's theorem that holds in this setting. A complete
characterization is given for the Euclidean space and Zoll manifolds (that is,
manifolds with periodic geodesic flow) via averaging formulae relating the
semiclassical measures corresponding to the evolution to those of the initial
states. The case of the flat torus is also addressed; it is shown that
non-classical behavior may occur when energy concentrates on resonant
frequencies. Moreover, we present an example showing that the semiclassical
measures associated to a sequence of states no longer determines those of their
evolutions. Finally, some results concerning the equation with a potential are
presented.Comment: 18 pages; Theorems 1,2 extendend to deal with arbitrary time-scales;
references adde
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