3,397 research outputs found
On the Drach superintegrable systems
Cubic invariants for two-dimensional degenerate Hamiltonian systems are
considered by using variables of separation of the associated St\"ackel
problems with quadratic integrals of motion. For the superintegrable St\"ackel
systems the cubic invariant is shown to admit new algebro-geometric
representation that is far more elementary than the all the known
representations in physical variables. A complete list of all known systems on
the plane which admit a cubic invariant is discussed.Comment: 16 pages, Latex2e+Amssym
Semiclassical Description of Wavepacket Revival
We test the ability of semiclassical theory to describe quantitatively the
revival of quantum wavepackets --a long time phenomena-- in the one dimensional
quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are
considered: time-dependent WKB and Van Vleck propagation. We show that both
approaches describe with impressive accuracy the autocorrelation function and
wavefunction up to times longer than the revival time. Moreover, in the Van
Vleck approach, we can show analytically that the range of agreement extends to
arbitrary long times.Comment: 10 pages, 6 figure
Differential constraints and exact solutions of nonlinear diffusion equations
The differential constraints are applied to obtain explicit solutions of
nonlinear diffusion equations. Certain linear determining equations with
parameters are used to find such differential constraints. They generalize the
determining equations used in the search for classical Lie symmetries
Exact Periodic Solutions of Shells Models of Turbulence
We derive exact analytical solutions of the GOY shell model of turbulence. In
the absence of forcing and viscosity we obtain closed form solutions in terms
of Jacobi elliptic functions. With three shells the model is integrable. In the
case of many shells, we derive exact recursion relations for the amplitudes of
the Jacobi functions relating the different shells and we obtain a Kolmogorov
solution in the limit of infinitely many shells. For the special case of six
and nine shells, these recursions relations are solved giving specific analytic
solutions. Some of these solutions are stable whereas others are unstable. All
our predictions are substantiated by numerical simulations of the GOY shell
model. From these simulations we also identify cases where the models exhibits
transitions to chaotic states lying on strange attractors or ergodic energy
surfaces.Comment: 25 pages, 7 figure
Long-Time Asymptotics of the Toda Lattice for Decaying Initial Data Revisited
The purpose of this article is to give a streamlined and self-contained
treatment of the long-time asymptotics of the Toda lattice for decaying initial
data in the soliton and in the similarity region via the method of nonlinear
steepest descent.Comment: 41 page
Angular distribution of photoluminescence as a probe of Bose Condensation of trapped excitons
Recent experiments on two-dimensional exciton systems have shown the excitons
collect in shallow in-plane traps. We find that Bose condensation in a trap
results in a dramatic change of the exciton photoluminescence (PL) angular
distribution. The long-range coherence of the condensed state gives rise to a
sharply focussed peak of radiation in the direction normal to the plane. By
comparing the PL profile with and without Bose Condensation we provide a simple
diagnostic for the existence of a Bose condensate. The PL peak has strong
temperature dependence due to the thermal order parameter phase fluctuations
across the system. The angular PL distribution can also be used for imaging
vortices in the trapped condensate. Vortex phase spatial variation leads to
destructive interference of PL radiation in certain directions, creating nodes
in the PL distribution that imprint the vortex configuration.Comment: 4 pages, 3 figure
Black Holes in Non-flat Backgrounds: the Schwarzschild Black Hole in the Einstein Universe
As an example of a black hole in a non-flat background a composite static
spacetime is constructed. It comprises a vacuum Schwarzschild spacetime for the
interior of the black hole across whose horizon it is matched on to the
spacetime of Vaidya representing a black hole in the background of the Einstein
universe. The scale length of the exterior sets a maximum to the black hole
mass. To obtain a non-singular exterior, the Vaidya metric is matched to an
Einstein universe. The behaviour of scalar waves is studied in this composite
model.Comment: 8 pages, 3 postscript figures, minor corrections Journal Ref:
accepted for Physical Review
Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites
We calculate the bulk contribution for the doubly degenerated largest
eigenvalue of the transfer matrix of the eight vertex model with an odd number
of lattice sites N in the disordered regime using the generic equation for
roots proposed by Fabricius and McCoy. We show as expected that in the
thermodynamic limit the result coincides with the one in the N even case.Comment: 11 pages LaTeX New introduction, Method change
Vortex Dynamics and Hall Conductivity of Hard Core Bosons
Magneto-transport of hard core bosons (HCB) is studied using an XXZ quantum
spin model representation, appropriately gauged on the torus to allow for an
external magnetic field. We find strong lattice effects near half filling. An
effective quantum mechanical description of the vortex degrees of freedom is
derived. Using semiclassical and numerical analysis we compute the vortex
hopping energy, which at half filling is close to magnitude of the boson
hopping energy. The critical quantum melting density of the vortex lattice is
estimated at 6.5x10-5 vortices per unit cell. The Hall conductance is computed
from the Chern numbers of the low energy eigenstates. At zero temperature, it
reverses sign abruptly at half filling. At precisely half filling, all
eigenstates are doubly degenerate for any odd number of flux quanta. We prove
the exact degeneracies on the torus by constructing an SU(2) algebra of
point-group symmetries, associated with the center of vorticity. This result is
interpreted as if each vortex carries an internal spin-half degree of freedom
('vspin'), which can manifest itself as a charge density modulation in its
core. Our findings suggest interesting experimental implications for vortex
motion of cold atoms in optical lattices, and magnet-transport of short
coherence length superconductors.Comment: 15 pages, 15 figure
The heat kernel for deformed spheres
We derive the asymptotic expansion of the heat kernel for a Laplace operator
acting on deformed spheres. We calculate the coefficients of the heat kernel
expansion on two- and three-dimensional deformed spheres as functions of
deformation parameters. We find that under some deformation the conformal
anomaly for free scalar fields on and is canceled.Comment: 10 pages, latex, no figure
- …