The theory of contractions of 2D 2nd order quantum superintegrable systems and its relation to the Askey scheme for hypergeometric orthogonal polynomials
Abstract
We describe a contraction theory for 2nd order superintegrable systems, showing that all such systems in 2 dimensions are limiting cases of a single system: the generic 3parameter potential on the 2sphere, in our listing. Analogously, all of the quadratic symmetry algebras of these systems can be obtained by a sequence of contractions starting from . By contracting function space realizations of irreducible representations of the algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.This relates the scheme directly to explicitly solvable quantum mechanical systems. Amazingly, all of these contractions of superintegrable systems with potential are uniquely induced by Wigner Lie algebra contractions of and . The present paper concentrates on describing this intimate link between Lie algebra and superintegrable system contractions, with the detailed calculations presented elsewhere. Joint work with E. Kalnins, S. Post, E. Subag and R. Heinonen.
1 Introduction
A quantum superintegrable system is an integrable Hamiltonian system on an dimensional Riemannian/pseudoRiemannian manifold with potential: , that admits algebraically independent partial differential operators commuting with , apparently the maximum possible.
Here, is the LaplaceBeltrami operator on the manifold. Superintegrability captures the properties of quantum Hamiltonian systems that allow the Schrödinger eigenvalue problem to be solved exactly, analytically and algebraically. There is a similar definition of classical superintegrable systems with Hamiltonian on phase space with functionally independent constants of the motion with and polynomial in the momenta, definitely the maximum number possible. A system is of order if the maximum order of the symmetry operators , other than , (or classically the maximum order of constants of the motion as polynomials) is . For , all systems are known. The symmetry operators of each system close under commutation (or under the Poisson bracket) to generate a quadratic algebra, and the irreducible representations of the algebra determine the eigenvalues of and their multiplicity. Classically we get important information about the orbits through algebraic methods alone. Detailed motivation for the study of superintegrable systems, a presentation of the theory and many references can be found in [16, 13]. All the 2nd order classical and quantum superintegrable systems are limiting cases of a single system: the generic 3parameter potential on the 2sphere, in our listing. Analogously all quadratic symmetry algebras of these systems are contractions of . In the quantum case this system is
(1) 
In the following sections we give brief descriptions of 1st and 2nd order 2D superintegrable systems, both free and with degenerate or nondegenerate potential. Every nonfree system is associated with a closed quadratic algebra generated by its symmetries. We state, and prove elsewhere, that a free system extends to a superintegrable system with potential if and only if its symmetries generate a closed free quadratic algebra. We point out that the theory of contractions of Lie symmetry algebras of constant curvature spaces is intimately associated with superintegrable systems of 1st order; indeed it appears to have been the motivation for the development of this theory by Wigner and Inönü. Then we show for systems on 2D constant curvature spaces how these Lie algebra contractions induce 1) contractions of the free quadratic algebras and then 2) induce contractions of the nondegenerate and degenerate quadratic algebras of systems with potential. Next we describe how the contractions of the superintegrable systems with potential can induce contractions of models of irreducible representations of the quadratic algebras through the process of ‘saving’ a representation. The Askey scheme for hypergeometric orthogonal polynomials emerges as a special subclass of these model contractions. We conclude with some observations.
2 1st and 2nd order 2D superintegrable systems
1st order systems : In the quantum case these are the (zeropotential) LaplaceBeltrami eigenvalue equations on constant curvature spaces, such as the Euclidean Helmholtz equation (or the KleinGordon equation ), and the Laplace Beltrami eigenvalue equation on the 2sphere The first order symmetries close under commutation to form the Lie algebras , or . The eigenspaces of these systems support differential operator models of the irreducible representations of the Lie algebras in which basis eigenfunctions are the spherical harmonics (),Bessel functions () and more complicated special functions [3, 5].
It was exactly these 1st order systems which motivated the pioneering work of Inönü and Wigner [2] on Lie algebra contractions. While, that paper introduced Lie algebra contractions in general, the motivation and virtually all the examples were of symmetry algebras of these systems. It was shown that contracts to . In the physical space this is accomplished by letting the radius of the sphere go to infinity, so that the surface flattens out. Under this limit the LaplaceBeltrami eigenvalue equation goes to the Helmholtz equation.
The following defines socalled natural contractions, [14], a generalization of WignerInönü contractions. Lie algebra contractions: Let , be two complex Lie algebras. We say is a contraction of if for every there exists a linear invertible map such that for every ,
Thus, as the 1parameter family of basis transformations can become nonsingular but the structure constants go to a finite limit.
Features of Wigner’s contraction approach, [2, 15]:

‘Saving’ a representation. Passing through a sequence of irreducible representations of the source symmetry algebra to obtain an irreducible representation of the target algebra in the contraction limit.

Simple models of irreducible representations. Finding models on function spaces so that the eigenfunctions of the generators are special functions.

Limit relations between special functions, as a result of saving a model representation in the contraction limit.

Use of the models to find expansion coefficients relating different special function bases.
Free 2nd order superintegrable systems in 2D: We will apply Wigner’s ideas to 2nd order systems in 2D . We start with the free (no potential function) case. The complex spaces with free Hamiltonians admitting at least three 2nd order symmetries (i.e., three 2nd order Killing tensors) were classified by Koenigs [11]. They are:

The two constant curvature spaces: flat space and the complex 2sphere. They each admit 6 linearly independent 2nd order symmetries and 3 1st order symmetries,

The four Darboux spaces, (4 2nd order symmetries and 1 1st order symmetry):

Eleven 4parameter Koenigs spaces (3 2nd order symmetries and no 1st order symmetries). An example is
2nd order superintegrable systems (with potential) in 2D: All such systems are known. There are 59 and each of the spaces classified by Koenigs admits at least one system. However, under the Stäckel transform, an invertible structure preserving mapping [13], the systems divide into 12 equivalence classes, each with a representative in a constant curvature space. Now the symmetry algebra is a quadratic algebra, not usually a Lie algebra, and the irreducible representations of the quantum algebra determine the eigenvalues of and their multiplicity
There are 3 types of superintegrable systems:

Nondegenerate: (3parameter potential)

Degenerate: (1parameter potential)

Free:
Usually the trivial added constant in each potential is ignored, though it is vital for some purposes.
Nondegenerate systems ( generators): The quantum symmetry algebra generated by always closes under commutation. Let be the 3rd order commutator of the generators. Then
Here is the symmetrizer of and . This structure is an example of a quadratic algebra. Here the are constants or polynomials in the parameters of the potential. The exact rules are given in [8] and [13].
Degenerate systems : There are 4 generators: one 1st order and 3 second order .
Since there must be an identity satisfied by the 4 generators. It is of 4th order:
Again the are constants or polynomials in the parameters of the potential.
The structure of classical quadratic algebras is similar, except no symmetrizers are needed. In [9] it is shown that all of the classical and quantum structure equations for nondegenerate systems can, in fact, be derived from the equation for , and all degenerate structure equations can be determined to within a constant factor from the 4th order identity.
Stäckel Equivalence Classes: There are 59 types of 2D 2nd order superintegrable systems, on a variety of manifolds but under the Stäckel transform, an invertible structure preserving mapping, they divide into 12 equivalence classes with representatives on flat space and the 2sphere, 6 with nondegenerate 3parameter potentials
and 6 with degenerate 1parameter potentials, [13],
The notation comes from [4] where all 2nd order 2D superintegrable systems on constant curvature spaces are classified.
3 Representatives of nondegenerate quantum systems

: Defined in (1).
Structure equations:

(WinternitzSmorodinsky system)
Generators:
Structure relations:

Generators:
Structure equations:
Here, the algebra generators are

Generators:
Structure relations:
The algebra generators are .

Generators:
Structure equations:
Here , , ,

Generators:
Structure relations:
Here, , , ,
4 Representatives of degenerate systems
There are close relations between nondegenerate and degenerate systems.

Every 1parameter potential can be obtained from some 3parameter potential by parameter restriction.

It is not simply a restriction, however, because the structure of the symmetry algebra changes.

A formally skewadjoint 1st order symmetry appears and this induces a new 2nd order symmetry.

Thus the restricted potential has a strictly larger symmetry algebra than is initially apparent.
We list the 6 representatives of the equivalence classes for degenerate systems:

(Higgs Oscillator)
The system is the same as with , with the former replaced by
and
Structure relations:

(Harmonic Oscillator)
Basis symmetries:
Also we set .
Structure equations:

Basis Symmetries: (with )
Structure equations:

Basis symmetries: (where )
Structure equations:

Basis symmetries: ()
Structure equations:

Basis symmetries: (with , )
Structure equations:
5 Contractions of superintegrable systems
Suppose we have a nondegenerate quantum superintegrable system with generators , and the usual structure equations, defining a quadratic algebra . If we make a change of basis to new generators and parameters such that
for some constant matrices such that , we will have the same system with new structure equations of the same form for , , , but with transformed structure constants.

Choose a continuous 1parameter family of basis transformation matrices , such that is the identity matrix, and , .

Now suppose as the basis change becomes singular, (i.e., the limits of either do not exist or, if they exist do not satisfy ) but the structure equations involving , go to a limit, defining a new quadratic algebra .

We call a contraction of in analogy with Lie algebra contractions.
There is a similar definition of a contraction of a degenerate superintegrable system. Further, we say that the 2D system without potential, , and with 3 algebraically independent secondorder symmetries is a 2nd order free triplet. The possible spaces admitting free triplets are just those classified by Koenigs. Note that every nondegenerate or degenerate superintegrable system defines a free triplet, simply by setting the parameters in the potential. Similarly, this free triplet defines a free quadratic algebra, i.e., a quadratic algebra with all . In general, a free triplet cannot be obtained as a restriction of a superintegrable system and its associated algebra does not close to a free quadratic algebra. All of these definitions extend easily to classical superintegrable systems.
We have the following closure theorems:
Theorem 1
Closure Theorem: A free triplet (classical or quantum) extends to a superintegrable system if and only if it generates a free quadratic algebra.
Theorem 2
A superintegrable system, degenerate or nondegenerate, is uniquely determined by its free quadratic algebra.
Proofs of these results will appear in [9]. The main ideas are as follows. Suppose we have a classical free triplet with basis
that determines a free nondegenerate quadratic algebra, hence a free nondegenerate superintegrable system. From the free system alone we can compute the functions , expressed in terms of the Cartesianlike coordinates , that determine the system of equations for an additive potential
(2) 
These equations always admit a constant potential for a solution, but they will admit a full 4dimensional vector space of solutions if and only if the integrability conditions for (2) are identically satisfied. In [9] we show that the integrability conditions hold if and only if the free system generates a quadratic algebra. This is an algebraic solution for an analytic problem. Further, if a potential function satisfies (2) then it is guaranteed that the BertrandDarboux integrability conditions for equations
hold and we can compute the solutions , , unique up to additive constants, such that the constants of the motion define a nondegenerate superintegrable system. This system is guaranteed to determine a nondegenerate quadratic algebra with potential whose highest order (potentialfree) terms agree with the free quadratic algebra. The functions are defined independent of the basis chosen for the free triplet although, of course, they do depend upon the particular coordinates chosen.
Similarly, there is an associated 2nd order quantum free triplet
that defines a free nondegenerate quantum quadratic algebra with potential. The functions are the same as before.
There is an analogous construction of degenerate superintegrable systems with potential from free triplets that generate a free quadratic algebras, but are such that one generator say, , is a perfect square.
6 Lie algebra contractions
The contractions of the Lie algebras and have long since been classified, e.g. [17]. There are 7 nontrivial contractions of and 4 of . However, 2 of the contractions of take it to an abelian Lie algebra so are not of interest to us.
WignerInonu contractions of :