arXiv subjects

Vincent X. Genest

Publications and source records attributed to Vincent X. Genest.

47 records · Page 3Linked to original sources

arXiv2013

The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states

An algebraic interpretation of the bivariate Krawtchouk polynomials is provided in the framework of the 3-dimensional isotropic harmonic oscillator model. These polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in 3 dimensions. Many of their properties are derived by exploiting the group-theoretic setting. The bivariate Tratnik polynomials of Krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters. It is explained how the approach generalizes naturally to (d+1) dimensions and allows to interpret multivariate Krawtchouk polynomials as matrix elements of SO(d+1) unitary representations. Indications are given on the connection with other algebraic models for these polynomials.

math-ph↗

arXiv2013

The Hahn superalgebra and supersymmetric Dunkl oscillator models

A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator model in the plane, whose Hamiltonian involves reflection operators. In this realization, the reflections act as grading operators and the odd generators are part of the Schwinger-Dunkl algebra, which is a two-parameter extension of the bosonic su(2) construction. The even part of the algebra is built from bilinears in the odd generators and satisfy the Hahn algebra supplemented with involutions. A family of supersymmetric Dunkl oscillator models in n dimensions is also considered. The Hamiltonians of these supersymmetric models differ from ordinary Dunkl oscillators by pure reflection terms. In two dimensions, the supersymmetric Dunkl oscillator is seen to have the even part of the Hahn superalgebra as invariance algebra.

math-ph↗

arXiv2013

Interbasis expansions for the isotropic 3D harmonic oscillator and bivariate Krawtchouk polynomials

An explicit expression for the general bivariate Krawtchouk polynomials is obtained in terms of the standard Krawtchouk and dual Hahn polynomials. The bivariate Krawtchouk polynomials occur as matrix elements of the unitary reducible representations of SO(3) on the energy eigenspaces of the 3-dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components. The decomposition entails expanding the Cartesian basis states in the spherical bases that span irreducible SO(3) representations. The overlap coefficients are obtained from the Clebsch-Gordan problem for the su(1,1) Lie algebra.

math-ph↗

arXiv2013

The equitable Racah algebra from three su(1,1) algebras

The Racah algebra, a quadratic algebra with two independent generators, is central in the analysis of superintegrable models and encodes the properties of the Racah polynomials. It is the algebraic structure behind the su(1,1) Racah problem as it is realized by the intermediate Casimir operators arising in the addition of three irreducible su(1,1) representations. It has been shown that this Racah algebra can also be obtained from quadratic elements in the enveloping algebra of su(2). The correspondence between these two realizations is here explained and made explicit.

math-ph↗

arXiv2013

The singular and the 2:1 anisotropic Dunkl oscillators in the plane

Two Dunkl oscillator models are considered: one singular and the other with a 2:1 frequency ratio. These models are defined by Hamiltonians which include the reflection operators in the two variables x and y. The singular or caged Dunkl oscillator is second-order superintegrable and admits separation of variables in both Cartesian and polar coordinates. The spectrum of the Hamiltonian is obtained algebraically and the separated wavefunctions are given in the terms of Jacobi, Laguerre and generalized Hermite polynomials. The symmetry generators are constructed from the su(1,1) dynamical operators of the one-dimensional model and generate a cubic symmetry algebra. In terms of the symmetries responsible for the separation of variables, the symmetry algebra of the singular Dunkl oscillator is quadratic and can be identified with a special case of the Askey-Wilson algebra AW(3) with central involutions. The 2:1 anisotropic Dunkl oscillator model is also second-order superintegrable. The energies of the system are obtained algebraically, the symmetry generators are constructed using the dynamical operators and the resulting symmetry algebra is quadratic. The general system appears to admit separation of variables only in Cartesian coordinates. Special cases where separation occurs in both Cartesian and parabolic coordinates are considered. In the latter case the wavefunctions satisfy the biconfluent Heun equation and depend on a transcendental separation constant.

math-ph↗

arXiv2013

Superintegrability in two dimensions and the Racah-Wilson algebra

The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere, is cast in the framework of the Racah problem for the su(1,1) algebra. The Hamiltonian of the 3-parameter system and the generators of its quadratic symmetry algebra are seen to correspond to the total and intermediate Casimir operators of the combination of three su(1,1) algebras, respectively. The construction makes explicit the isomorphism between the Racah-Wilson algebra, which is the fundamental algebraic structure behind the Racah problem for su(1,1), and the invariance algebra of the generic 3-parameter system. It also provides an explanation for the occurrence of the Racah polynomials as overlap coefficients in this context. The irreducible representations of the Racah-Wilson algebra are reviewed as well as their connection with the Askey scheme of classical orthogonal polynomials.

math-ph↗

arXiv2013

Bispectrality of the Complementary Bannai-Ito Polynomials

A one-parameter family of operators that have the complementary Bannai-Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai-Ito polynomials and also correspond to a $q\rightarrow-1$ limit of the Askey-Wilson polynomials. The eigenvalue equations for the CBI polynomials are found to involve second order Dunkl shift operators with reflections and exhibit quadratic spectra. The algebra associated to the CBI polynomials is given and seen to be a deformation of the Askey-Wilson algebra with an involution. The relation between the CBI polynomials and the recently discovered dual -1 Hahn and para-Krawtchouk polynomials, as well as their relation with the symmetric Hahn polynomials, is also discussed.

math.CA↗

arXiv2013

The Dunkl oscillator in the plane II : representations of the symmetry algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger-Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra u(2). Two of the symmetry generators, J_3 and J_2, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J_3 is diagonal and the operator J_2 acts in a tridiagonal fashion. In the circular basis, the operator J_2 is block upper-triangular with all blocks 2x2 and the operator J_3 acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J_2 in the circular basis are generated by the Heun polynomials and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J_2 are generated by little -1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J_2 is diagonal is then considered. In this basis, the defining relations of the Schwinger-Dunkl algebra imply that J_3 acts in a block tridiagonal fashion with all blocks 2x2. The matrix elements of J_3 in this basis are given explicitly.

math-ph↗

arXiv2013

The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra generated by the constants of motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie algebra u(2) with involutions. The system admits separation of variables in both Cartesian and polar coordinates. The separated wavefunctions are respectively expressed in terms of generalized Hermite polynomials and products of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi-Dunkl polynomials appear as eigenfunctions of the symmetry operator responsible for the separation of variables in polar coordinates. The expansion coefficients between the Cartesian and polar bases (overlap coefficients) are given as linear combinations of dual -1 Hahn polynomials. The connection with the Clebsch-Gordan problem of the sl_{-1}(2) algebra is explained.

math-ph↗

arXiv2012

Generalized squeezed-coherent states of the finite one-dimensional oscillator and matrix multi-orthogonality

A set of generalized squeezed-coherent states for the finite u(2) oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the su(2) generators. These matrix elements are given in the (N+1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed-coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2x2 matrix orthogonal polynomials and the squeezed-coherent states tend to those of the standard oscillator.

math-ph↗

arXiv2012

d-Orthogonal polynomials and su(2)

Two families of d-orthogonal polynomials related to su(2) are identified and studied. The algebraic setting allows their full characterization (explicit expressions, recurrence relations, difference equations, generating functions, etc.) of those polynomials. In the limit where su(2) contracts to the Heisenberg-Weyl algebra h_1, these polynomials tend to the standard Meixner polynomials and d-Charlier polynomials, respectively.

math-ph↗