489 research outputs found
Complete sets of invariants for dynamical systems that admit a separation of variables
Consider a classical Hamiltonian H in n dimensions consisting of a kinetic energy term plus a potential. If the associated HamiltonâJacobi equation admits an orthogonal separation of variables, then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, ,Pn are the other second-order constants of the motion associated with the separable coordinates, and {Qi,Qj} = {Pi,Pj} = 0, {Qi,Pj} = ij. The 2nâ1 functions Q2, ,Qn,P1, ,Pn form a basis for the invariants. We show how to determine for exactly which spaces and potentials the invariant Qj is a polynomial in the original momenta. We shed light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. For n = 2 we go further and consider all cases where the HamiltonâJacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion
Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV
This is the second paper on the path integral approach of superintegrable
systems on Darboux spaces, spaces of non-constant curvature. We analyze in the
spaces \DIII and \DIV five respectively four superintegrable potentials,
which were first given by Kalnins et al. We are able to evaluate the path
integral in most of the separating coordinate systems, leading to expressions
for the Green functions, the discrete and continuous wave-functions, and the
discrete energy-spectra. In some cases, however, the discrete spectrum cannot
be stated explicitly, because it is determined by a higher order polynomial
equation.
We show that also the free motion in Darboux space of type III can contain
bound states, provided the boundary conditions are appropriate. We state the
energy spectrum and the wave-functions, respectively
Superintegrable systems with spin and second-order integrals of motion
We investigate a quantum nonrelativistic system describing the interaction of
two particles with spin 1/2 and spin 0, respectively. We assume that the
Hamiltonian is rotationally invariant and parity conserving and identify all
such systems which allow additional integrals of motion that are second order
matrix polynomials in the momenta. These integrals are assumed to be scalars,
pseudoscalars, vectors or axial vectors. Among the superintegrable systems
obtained, we mention a generalization of the Coulomb potential with scalar
potential and spin orbital one
.Comment: 32 page
Third order superintegrable systems separating in polar coordinates
A complete classification is presented of quantum and classical
superintegrable systems in that allow the separation of variables in
polar coordinates and admit an additional integral of motion of order three in
the momentum. New quantum superintegrable systems are discovered for which the
potential is expressed in terms of the sixth Painlev\'e transcendent or in
terms of the Weierstrass elliptic function
Structure results for higher order symmetry algebras of 2D classical superintegrable systems
Recently the authors and J.M. Kress presented a special function recurrence
relation method to prove quantum superintegrability of an integrable 2D system
that included explicit constructions of higher order symmetries and the
structure relations for the closed algebra generated by these symmetries. We
applied the method to 5 families of systems, each depending on a rational
parameter k, including most notably the caged anisotropic oscillator, the
Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system.
Here we work out the analogs of these constructions for all of the associated
classical Hamiltonian systems, as well as for a family including the generic
potential on the 2-sphere. We do not have a proof in every case that the
generating symmetries are of lowest possible order, but we believe this to be
so via an extension of our method.Comment: 23 page
Focusing a fountain of neutral cesium atoms with an electrostatic lens triplet
An electrostatic lens with three focusing elements in an alternating-gradient
configuration is used to focus a fountain of cesium atoms in their ground
(strong-field-seeking) state. The lens electrodes are shaped to produce only
sextupole plus dipole equipotentials which avoids adding the unnecessary
nonlinear forces present in cylindrical lenses. Defocusing between lenses is
greatly reduced by having all of the main electric fields point in the same
direction and be of nearly equal magnitude. The addition of the third lens gave
us better control of the focusing strength in the two transverse planes and
allowed focusing of the beam to half the image size in both planes. The beam
envelope was calculated for lens voltages selected to produced specific
focusing properties. The calculations, starting from first principles, were
compared with measured beam sizes and found to be in good agreement.
Application to fountain experiments, atomic clocks, and focusing polar
molecules in strong-field-seeking states is discussed.Comment: 8 pages 10 figure
Some Spacetimes with Higher Rank Killing-Stackel Tensors
By applying the lightlike Eisenhart lift to several known examples of
low-dimensional integrable systems admitting integrals of motion of
higher-order in momenta, we obtain four- and higher-dimensional Lorentzian
spacetimes with irreducible higher-rank Killing tensors. Such metrics, we
believe, are first examples of spacetimes admitting higher-rank Killing
tensors. Included in our examples is a four-dimensional supersymmetric pp-wave
spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy
a non-trivial Poisson-Schouten-Nijenhuis algebra. We discuss the extension to
the quantum regime
Louisville Ridge subduction at the Tonga-Kermadec trench: preliminary velocity models from wide-angle seismics
The Coulomb-Oscillator Relation on n-Dimensional Spheres and Hyperboloids
In this paper we establish a relation between Coulomb and oscillator systems
on -dimensional spheres and hyperboloids for . We show that, as in
Euclidean space, the quasiradial equation for the dimensional Coulomb
problem coincides with the -dimensional quasiradial oscillator equation on
spheres and hyperboloids. Using the solution of the Schr\"odinger equation for
the oscillator system, we construct the energy spectrum and wave functions for
the Coulomb problem.Comment: 15 pages, LaTe
Superintegrability and higher order polynomial algebras II
In an earlier article, we presented a method to obtain integrals of motion
and polynomial algebras for a class of two-dimensional superintegrable systems
from creation and annihilation operators. We discuss the general case and
present its polynomial algebra. We will show how this polynomial algebra can be
directly realized as a deformed oscillator algebra. This particular algebraic
structure allows to find the unitary representations and the corresponding
energy spectrum. We apply this construction to a family of caged anisotropic
oscillators. The method can be used to generate new superintegrable systems
with higher order integrals. We obtain new superintegrable systems involving
the fourth Painleve transcendent and present their integrals of motion and
polynomial algebras.Comment: 11 page
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