A superintegrable finite oscillator in two dimensions with symmetry
A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials form bases for irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families are related to each other by an automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points goes to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials.
There is a considerable body of knowledge on superintegrable models and remarkably, the recent years have witnessed significant advances in their classification. The harmonic oscillator is the prototype of superintegrable models. These are Hamiltonian systems which admit a number of conserved quantities greater than the dimension in which they are defined. They are called maximally superintegrable when this number is . In quantum mechanics, these constants of motion are typically required to be algebraically independent. To a large extent, the documented cases describe continuous systems and little is known in the discrete realm. It is the purpose of the present paper to present a finite analog of the quantum harmonic oscillator in two dimensions which is maximally superintegrable.
There is by now a number of finite models of the quantum oscillator in one dimension. The most studied is based on the algebra . Generalizations that model the parabosonic oscillator have been recently developed with extended algebras [13, 14, 15]. Let us recall the main features of the system associated with the algebra whose generators satisfy and and whose standard -dimensional irreducible representations may be taken to have basis vectors , , defined by with a non-negative integer related to the value of the Casimir operator. Upon making the identification
the commutation relations are interpreted as the Heisenberg equations of motion of the oscillator:
The wavefunctions of this finite oscillator are, naturally, the overlaps between the eigenstates of the position operator and those of , ie. ; they are hence given by rotation matrix elements known to be expressible in terms of the Krawtchouk polynomials  with and , the eigenvalues of . The Krawtchouk polynomials are given in terms of a hypergeometric series
and are orthogonal with respect to the binomial distribution
This entails a realization of the observables of this finite oscillator in terms of finite difference operators. It is known that contracts to the Heisenberg algebra. Hence, when and with proper rescaling, the description of the standard (continuous) harmonic oscillator is recovered  as the Krawtchouk polynomials tend to the Hermite polynomials in the limit .
The Krawtchouk (or ) one-dimensional finite oscillator has been used to build two-dimensional models. A first approach was to take the direct product of two finite one-dimensional oscillators to obtain a two-dimensional system defined on a square grid of points and with as dynamical algebra. A second one  was to use the isomorphism between and to describe this finite two-dimensional oscillator in terms of discrete radial and angular coordinates. As remarked in Refs. [1, 4, 5], these models do not possess the symmetry algebra of the continuous two-dimensional quantum oscillator. This symmetry algebra (not to be confused with the dynamical algebras) is generated by the three constants of motion which make the continuous harmonic oscillator maximally superintegrable. The two-dimensional finite models constructed so far do not therefore share that property. We here introduce a system that has such an symmetry. The model will be defined on a grid of triangular shape.
Finite oscillator models have various uses e.g. in optical image processing and in signal analysis where only a finite number of eigenvalues exist [2, 3]. Finite planar oscillators can be employed in particular to describe waveguides and pixellated screens. It is expected that preserving, at the discrete level, all the symmetries of the continuous oscillator model would prove advantageous in these applications.
It is natural to think that two-variable generalizations of the Krawtchouk polynomials could provide an underpinning for interesting two-dimensional oscillator models. As a matter of fact, two families of such bivariate polynomials have been identified. Some 20 years ago, Tratnik introduced multivariable Racah polynomials depending on two parameters, that encompass multivariable Krawtchouk polynomials in their simplest case. Geronimo and Iliev  subsequently showed that these polynomials are bispectral, i.e. that they obey, in the bivariate case, a pair of difference equations in addition to the recurrence relations associated with their orthogonality. Griffiths  in the early 70’s and more recently Hoare and Rahman  discussed a second family of bivariate Krawtchouk polynomials with four parameters in connection with a probabilistic model. (See also Ref. .) These were later extended to an arbitrary number of variables  and shown to be also bispectral . In contradistinction with the Tratnik version, the Krawtchouk polynomials of Rahman satisfy a nearest neighbor five-term recurrence relation . They have been used to construct a spin lattice with remarkable quantum state transfer properties .
In view of the bispectrality of the two-variable Krawtchouk polynomials, it is of interest to inquire what is the quantum mechanics that their difference equations describe and to find the corresponding continuous dynamics in the limit . This last question involves obtaining the limit of the bivariate Krawtchouk polynomials, an issue that seems not to have been resolved so far.
The Rahman polynomials lead to the most direct interpretation. Remarkably, their limit is found to be the product of two Hermite polynomials. This is obtained by observing that their difference equations factorize in terms of a pair of simple creation and annihilation operators which obey Heisenberg commutation relations. A finite model for the two-dimensional isotropic (or anisotropic) oscillator can then be constructed in the standard fashion. In terms of these creation and annihilation operators, the constants of motion take the same form as in the continuous situation and the symmetry algebra is therefore preserved in the discretization process. For the isotropic case, the Rahman polynomials are shown to form a basis for the irreducible representation spaces of They correspond in the continuum limit to wave functions that are separated in Cartesian coordinates. In view of the eigenvalue equations that they obey , the Tratnik polynomials are seen to correspond to separation of variables in coordinates that are rotated relative to the ones associated with the Rahman polynomials. This is why the Tratnik polynomials depend essentially on one less parameter than those of Rahman; the additional parameter has been removed by the rotation. It should be stressed that the oscillator models based on these bivariate Krawtchouk polynomials are defined on a uniform grid of triangular shape.
The remainder of the paper is organized as follows. In section 2, we present an invariant finite isotropic oscillator model, whose eigenfunctions are given in terms of the two-variable Krawtchouk polynomials of Hoare and Rahman, and we describe its continuum limit. Also, a finite anisotropic oscillator is introduced which is seen to be superintegrable for rational frequencies of the coupling constants. In section 3, we show how the two-variable Krawtchouk polynomials of Tratnik can also be realized as eigenfunctions of the same isotropic Hamiltonian, though associated with a different integral of motion. Conclusions and an Appendix on the large limit of the trinomial distribution follow.
2 Finite oscillator model and the two-variable Krawtchouk polynomials of Hoare and Rahman
It should be noted that the polynomials depend essentially on three independent parameters. This fact can be observed in terms of the by writing
and equivalently for and .
With and nonnegative integers such that , the polynomials are defined on a uniform lattice of a triangular shape, i.e. and are orthogonal with respect to the two-variable generalization of the weight (4) given by the trinomial distribution
It should be noted that the constants satisfy the following three functional relations :
which also imply that the polynomials depend essentially on three parameters.
One of the specific features which the polynomials possess is that they obey a nearest neighbor difference equation in  (and also some contiguity relations in because of their duality):
where is a difference operator defined by . Recently, it was shown [12, 10] that the polynomials additionally satisfy a seven point difference equation, which is equivalent to the fact that the polynomials are eigenfunctions of the following difference operators:
The eigenvalue equations read:
As it will later become important, the sum depends only on the two parameters and .
A different combination of the operators (12) eliminates the non-nearest neighbor terms involving and the difference equation (11) recovered. Moreover, one can easily find that and give the “recurrence” relations for the polynomials :
If we set when or is a negative integer and given that at from (5), we can determine all the from (14). In that sense, the operators and fully characterize the Rahman polynomials defined on the grid points such that as their joint eigenfunctions.
2.1 The invariant isotropic Hamiltonian
In this section, we first consider the operators defined by
where is the shift operator defined by . One can easily verify that these operators satisfy a shifted form of the Heisenberg algebra relations:
and provide the factorizations of the operators and :
Let us consider the polynomials . It is straightforward to check from (16) that
This means that are also the Rahman polynomials up to a multiplicative constant, which is determined by evaluating at . We can thus obtain the following four relations:
Now let us go back to the relations (16). From these, it is clear that the Hamiltonian
admits the following integrals of the motion
The conserved quantities (21) form a basis for an algebra as they satisfy
and hence the Hamiltonian of (20) is invariant. It is readily seen that the Casimir operator
takes the form
It hence follows that the Rahman polynomials (5) with fixed form a basis for the -dimensional irreducible representations of . They diagonalize the operators or equivalently the commuting pair and with eigenvalues
2.2 An anisotropic Hamiltonian
as Hamiltonians will be integrable for arbitrary and and superintegrable when the ratio of the frequencies is rational. In each case, the eigenfunctions for the pair and either or are the Rahman polynomials. It is important to note that whereas depends essentially only on two parameters and , the anisotropic Hamiltonian depends on all three functionally independent parameters.
2.3 Continuum limit
In this subsection, we consider the limit of the two-variable Krawtchouk polynomials (5). The limit will be obtained by the following change of variables
Note that as , the range of and becomes the whole real line.
The operators have the following limits
Thus, in the limit , the Hamiltonian given in (20) tends to the Hamiltonian of a two-dimensional oscillator
which has been conjugated by the ground state and with the coupling constants absorbed into the variables.
While the operators and do not change the quantum numbers and , the ladder operators do and so the the normalization of the polynomials (see below) affects the limit of these operators. In order to find the proper choice of normalization, note that in the large limit the trinomial distribution tends to the two variable Gaussian (for a proof see A) with the following coefficients
Note that the constants in the change of variables (29) satisfy
and so the unit of area, as and are shifted by 1, is Thus,
The normalization of the Rahman polynomials given in  is
This suggests the following redefinition:
With this normalization, the orthogonality relation becomes
which implies, in the limit
Thus, the functions satisfy the eigenvalue equation
and a similar one in . They are moreover square integrable in the plane with respect to the measure and are hence polynomial. The normalization constant has been chosen to agree with a product of Hermite polynomials and so the limit satisfies
Using the proper normalization, the action of the ladder operators becomes
and similarly for the operators As expected, the limit of these scaled operators becomes the ladder operators for the Hermite polynomials
3 Two-variable Krawtchouk polynomials of Tratnik
A different set of commuting operators in the enveloping algebra of (15) lead to the version of two-variable Krawtchouk polynomials defined by Tratnik [19, 20]. Beginning with the observation that the isotropic Hamiltonian depends only on the two parameters and , it is also possible to construct another operator written in terms of the creation and annihilation operators (15), given (17), that depends only on these two parameters. The set of commuting operators is given by