181 research outputs found
Localized spectral asymptotics for boundary value problems and correlation effects in the free Fermi gas in general domains
We rigorously derive explicit formulae for the pair correlation function of
the ground state of the free Fermi gas in the thermodynamic limit for general
geometries of the macroscopic regions occupied by the particles and arbitrary
dimension. As a consequence we also establish the asymptotic validity of the
local density approximation for the corresponding exchange energy. At constant
density these formulae are universal and do not depend on the geometry of the
underlying macroscopic domain. In order to identify the correlation effects in
the thermodynamic limit, we prove a local Weyl law for the spectral asymptotics
of the Laplacian for certain quantum observables which are themselves dependent
on a small parameter under very general boundary conditions
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
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
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
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
Delocalization of slowly damped eigenmodes on Anosov manifolds
We look at the properties of high frequency eigenmodes for the damped wave
equation on a compact manifold with an Anosov geodesic flow. We study
eigenmodes with spectral parameters which are asymptotically close enough to
the real axis. We prove that such modes cannot be completely localized on
subsets satisfying a condition of negative topological pressure. As an
application, one can deduce the existence of a "strip" of logarithmic size
without eigenvalues below the real axis under this dynamical assumption on the
set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two
reference
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
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
Quantum statistical information contained in a semi-classical Fisher--Husimi measure
We study here the difference between quantum statistical treatments and
semi-classical ones, using as the main research tool a semi-classical,
shift-invariant Fisher information measure built up with Husimi distributions.
Its semi-classical character notwithstanding, this measure also contains
information of a purely quantal nature.
Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies
and to discover thermodynamic-like relations that involve the degree of
delocalization. Fisher-related thermal uncertainty relations are developed and
the degree of purity of canonical distributions, regarded as mixed states, is
connected to this Fisher measure as well.Comment: 9 pages, 3 figures; chenged conten
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