172 research outputs found
Phase Space Quantum Mechanics on the Anti-De Sitter Spacetime and its Poincar\'e Contraction
In this work we propose an alternative description of the quantum mechanics
of a massive and spinning free particle in anti-de~Sitter spacetime, using a
phase space rather than a spacetime representation. The regularizing character
of the curvature appears clearly in connection with a notion of localization in
phase space which is shown to disappear in the zero curvature limit. We show in
particular how the anti-de~Sitter optimally localized (coherent) states
contract to plane waves as the curvature goes to zero. In the first part we
give a detailed description of the classical theory {\it \a la Souriau\/}. This
serves as a basis for the quantum theory which is constructed in the second
part using methods of geometric quantization. The invariant positive K\"ahler
polarization that selects the anti-de~Sitter quantum elementary system is shown
to have as zero curvature limit the Poincar\'e polarization which is no longer
K\"ahler. This phenomenon is then related to the disappearance of the notion of
localization in the zero curvature limit.Comment: 37 pgs+3 figures (not included), PlainTeX, Preprint CRM-183
Modulational instability in dispersion-kicked optical fibers
We study, both theoretically and experimentally, modulational instability in
optical fibers that have a longitudinal evolution of their dispersion in the
form of a Dirac delta comb. By means of Floquet theory, we obtain an exact
expression for the position of the gain bands, and we provide simple analytical
estimates of the gain and of the bandwidths of those sidebands. An experimental
validation of those results has been realized in several microstructured fibers
specifically manufactured for that purpose. The dispersion landscape of those
fibers is a comb of Gaussian pulses having widths much shorter than the period,
which therefore approximate the ideal Dirac comb. Experimental spontaneous MI
spectra recorded under quasi continuous wave excitation are in good agreement
with the theory and with numerical simulations based on the generalized
nonlinear Schr\"odinger equation
Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries
Devices exhibiting the integer quantum Hall effect can be modeled by
one-electron Schroedinger operators describing the planar motion of an electron
in a perpendicular, constant magnetic field, and under the influence of an
electrostatic potential. The electron motion is confined to unbounded subsets
of the plane by confining potential barriers. The edges of the confining
potential barrier create edge currents. In this, the first of two papers, we
prove explicit lower bounds on the edge currents associated with one-edge,
unbounded geometries formed by various confining potentials. This work extends
some known results that we review. The edge currents are carried by states with
energy localized between any two Landau levels. These one-edge geometries
describe the electron confined to certain unbounded regions in the plane
obtained by deforming half-plane regions. We prove that the currents are stable
under various potential perturbations, provided the perturbations are suitably
small relative to the magnetic field strength, including perturbations by
random potentials. For these cases of one-edge geometries, the existence of,
and the estimates on, the edge currents imply that the corresponding
Hamiltonian has intervals of absolutely continuous spectrum. In the second
paper of this series, we consider the edge currents associated with two-edge
geometries describing bounded, cylinder-like regions, and unbounded,
strip-like, regions.Comment: 68 page
QUANTIZATION OF A CLASS OF PIECEWISE AFFINE TRANSFORMATIONS ON THE TORUS
We present a unified framework for the quantization of a family of discrete
dynamical systems of varying degrees of "chaoticity". The systems to be
quantized are piecewise affine maps on the two-torus, viewed as phase space,
and include the automorphisms, translations and skew translations. We then
treat some discontinuous transformations such as the Baker map and the
sawtooth-like maps. Our approach extends some ideas from geometric quantization
and it is both conceptually and calculationally simple.Comment: no. 28 pages in AMSTE
Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases
We prove approach to thermal equilibrium for the fully Hamiltonian dynamics
of a dynamical Lorentz gas, by which we mean an ensemble of particles moving
through a -dimensional array of fixed soft scatterers that each possess an
internal harmonic or anharmonic degree of freedom to which moving particles
locally couple. We establish that the momentum distribution of the moving
particles approaches a Maxwell-Boltzmann distribution at a certain temperature
, provided that they are initially fast and the scatterers are in a
sufficiently energetic but otherwise arbitrary stationary state of their free
dynamics--they need not be in a state of thermal equilibrium. The temperature
to which the particles equilibrate obeys a generalized equipartition
relation, in which the associated thermal energy is equal to
an appropriately defined average of the scatterers' kinetic energy. In the
equilibrated state, particle motion is diffusive
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