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Second Order Superintegrable Systems in Three Dimensions
A classical (or quantum) superintegrable system on an n-dimensional
Riemannian manifold is an integrable Hamiltonian system with potential that
admits 2n-1 functionally independent constants of the motion that are
polynomial in the momenta, the maximum number possible. If these constants of
the motion are all quadratic, the system is second order superintegrable. Such
systems have remarkable properties. Typical properties are that 1) they are
integrable in multiple ways and comparison of ways of integration leads to new
facts about the systems, 2) they are multiseparable, 3) the second order
symmetries generate a closed quadratic algebra and in the quantum case the
representation theory of the quadratic algebra yields important facts about the
spectral resolution of the Schr\"odinger operator and the other symmetry
operators, and 4) there are deep connections with expansion formulas relating
classes of special functions and with the theory of Exact and Quasi-exactly
Solvable systems. For n=2 the author, E.G. Kalnins and J. Kress, have worked
out the structure of these systems and classified all of the possible spaces
and potentials. Here I discuss our recent work and announce new results for the
much more difficult case n=3.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Lie theory and difference equations. I
AbstractA factorization method is constructed for sequences of second-order linear difference equations in analogy with the factorization method for differential equations. Six factorization types are established and recursion relations are obtained for various classes of special functions, among which are the hypergeometric functions and their limits, and the classical polynomials of a discrete variable: Tchebycheff, Krawtchouk, Charlier, Meixner, and Hahn. It is shown that the factorization method is a disguised form of Lie algebra representation theory
Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
We show that the symmetry operators for the quantum superintegrable system on
the 3-sphere with generic 4-parameter potential form a closed quadratic algebra
with 6 linearly independent generators that closes at order 6 (as differential
operators). Further there is an algebraic relation at order 8 expressing the
fact that there are only 5 algebraically independent generators. We work out
the details of modeling physically relevant irreducible representations of the
quadratic algebra in terms of divided difference operators in two variables. We
determine several ON bases for this model including spherical and cylindrical
bases. These bases are expressed in terms of two variable Wilson and Racah
polynomials with arbitrary parameters, as defined by Tratnik. The generators
for the quadratic algebra are expressed in terms of recurrence operators for
the one-variable Wilson polynomials. The quadratic algebra structure breaks the
degeneracy of the space of these polynomials. In an earlier paper the authors
found a similar characterization of one variable Wilson and Racah polynomials
in terms of irreducible representations of the quadratic algebra for the
quantum superintegrable system on the 2-sphere with generic 3-parameter
potential. This indicates a general relationship between 2nd order
superintegrable systems and discrete orthogonal polynomials
Contractions of Degenerate Quadratic Algebras, Abstract and Geometric
Quadratic algebras are generalizations of Lie algebras which 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 B\^ocher contractions of the conformal Lie algebra to itself. In 2 dimensions there are two kinds of quadratic algebras,
nondegenerate and degenerate. In the geometric case these correspond to 3
parameter and 1 parameter potentials, respectively. In a previous paper we
classified all abstract parameter-free nondegenerate quadratic algebras in
terms of canonical forms and determined which of these can be realized as
quadratic algebras of 2D nondegenerate superintegrable systems on constant
curvature spaces and Darboux spaces, and studied the relationship between
B\^ocher contractions of these systems and abstract contractions of the free
quadratic algebras. Here we carry out an analogous study of abstract
parameter-free degenerate quadratic algebras and their possible geometric
realizations. We show that the only free degenerate quadratic algebras that can
be constructed in phase space are those that arise from superintegrability. We
classify all B\^ocher contractions relating degenerate superintegrable systems
and, separately, all abstract contractions relating free degenerate quadratic
algebras. We point out the few exceptions where abstract contractions cannot be
realized by the geometric B\^ocher contractions
Separation of variables and the XXZ Gaudin magnet
In this work we generalise previous results connecting (rational) Gaudin
magnet models and classical separation of variables. It is shown that the
connection persists for the case of linear r-matrix algebra which corresponds
to the trigonometric 4x4 r-matrix (of the XXZ type). We comment also on the
corresponding problem for the elliptic (XYZ) r-matrix. A prescription for
obtaining integrable systems associated with multiple poles of an L-operator is
given.Comment: 11 pages, AMS-Te
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