15 research outputs found
Tools for Verifying Classical and Quantum Superintegrability
Recently many new classes of integrable systems in n dimensions occurring in
classical and quantum mechanics have been shown to admit a functionally
independent set of 2n-1 symmetries polynomial in the canonical momenta, so that
they are in fact superintegrable. These newly discovered systems are all
separable in some coordinate system and, typically, they depend on one or more
parameters in such a way that the system is superintegrable exactly when some
of the parameters are rational numbers. Most of the constructions to date are
for n=2 but cases where n>2 are multiplying rapidly. In this article we
organize a large class of such systems, many new, and emphasize the underlying
mechanisms which enable this phenomena to occur and to prove
superintegrability. In addition to proofs of classical superintegrability we
show that the 2D caged anisotropic oscillator and a Stackel transformed version
on the 2-sheet hyperboloid are quantum superintegrable for all rational
relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum
superintegrable for all rational values of a parameter k in the potential
Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
Two-dimensional quadratic algebras are generalizations of Lie algebras that
include the symmetry algebras of 2nd order superintegrable systems in 2
dimensions as special cases. The superintegrable systems are exactly solvable
physical systems in classical and quantum mechanics. Distinct superintegrable
systems and their quadratic algebras can be related by geometric contractions,
induced by In\"on\"u-Wigner type Lie algebra contractions. These geometric
contractions have important physical and geometric meanings, such as obtaining
classical phenomena as limits of quantum phenomena as and
nonrelativistic phenomena from special relativistic as , and the
derivation of the Askey scheme for obtaining all hypergeometric orthogonal
polynomials as limits of Racah/Wilson polynomials. In this paper we show how to
simplify the structure relations for abstract nondegenerate and degenerate
quadratic algebras and their contractions. In earlier papers we have classified
contractions of 2nd order superintegrable systems on constant curvature spaces
and have shown that all results are derivable from free quadratic algebras
contained in the enveloping algebras of the Lie algebras in
flat space and on nonzero constant curvature spaces. The
quadratic algebra contractions are induced by generalizations of
In\"on\"u-Wigner contractions of these Lie algebras. As a special case we
obtained the Askey scheme for hypergeometric orthogonal polynomials. Here we
complete this theoretical development for 2D superintegrable systems by showing
that the Darboux superintegrable systems are also characterized by free
quadratic algebras contained in the symmetry algebras of these spaces and that
their contractions are also induced by In\"on\"u-Wigner contractions. We
present tables of the contraction results
Special functions lie theory and partial differential equations
In the lectures an outline is given of the close relationship between special functions and Lie group theory. In doing so the exposition is deliberately quite direct and hopefully clear. An attempt is made to give an outline of the state of the subject if only from the author's point of view. Principal references containing the important features of the subject are also given
Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
There are 13 equivalence classes of 2D second order quantum and classical
superintegrable systems with nontrivial potential, each associated with a
quadratic algebra of hidden symmetries. We study the finite and infinite
irreducible representations of the quantum quadratic algebras though the
construction of models in which the symmetries act on spaces of functions of a
single complex variable via either differential operators or difference
operators. In another paper we have already carried out parts of this analysis
for the generic nondegenerate superintegrable system on the complex 2-sphere.
Here we carry it out for a degenerate superintegrable system on the 2-sphere.
We point out the connection between our results and a position dependent mass
Hamiltonian studied by Quesne. We also show how to derive simple models of the
classical quadratic algebras for superintegrable systems and then obtain the
quantum models from the classical models, even though the classical and quantum
quadratic algebras are distinct.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials
The structure theory for the quadratic algebra generated by first and second
order constants of the motion for 2D second order superintegrable systems with
nondegenerate (3-parameter) and or 2-parameter potentials is well understood,
but the results for the strictly 1-parameter case have been incomplete. Here we
work out this structure theory and prove that the quadratic algebra generated
by first and second order constants of the motion for systems with 4 second
order constants of the motion must close at order three with the functional
relationship between the 4 generators of order four. We also show that every
1-parameter superintegrable system is St\"ackel equivalent to a system on a
constant curvature space