Roy Oste

Generalized Fourier Transforms Arising from the Enveloping Algebras of (2) and (1∣2)

The Howe dual pair (sl(2), O(m)) allows the characterization of the classical Fourier transform (FT) on the space of rapidly decreasing functions as the exponential of a well-chosen element of sl(2) such that the Helmholtz relations are satisfied. In this paper, we first investigate what happens when instead we consider exponentials of elements of the universal enveloping algebra of sl(2). This leads to a complete class of generalized FTs, that all satisfy properties similar to the classical FT. There is moreover a finite subset of transforms which very closely resemble the FT. We obtain operator exponential expressions for all these transforms by making extensive use of the theory of integer-valued polynomials. We also find a plane wave decomposition of their integral kernel and establish uncertainty principles. In important special cases we even obtain closed formulas for the integral kernels. In the second part of the paper, the same problem is considered for the dual pair (osp(1 vertical bar 2), Spin(m)), in the context of the Dirac operator on R-m. This connects our results with the Clifford-FT studied in previous work.

Doubling (dual) Hahn polynomials : classification and applications

We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov El., Stoilova N.J., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christof fel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.

Tridiagonal test matrices for eigenvalue computations

The Clement or SylvesterKac matrix is a tridiagonal matrix with zero diagonal and simple integer entries. Its spectrum is known explicitly and consists of integers which makes it a useful test matrix for numerical eigenvalue computations. We consider a new class of appealing two-parameter extensions of this matrix which have the same simple structure and whose eigenvalues are also given explicitly by a simple closed form expression. The aim of this paper is to present in an accessible form these new matrices and examine some numerical results regarding the use of these extensions as test matrices for numerical eigenvalue computations.

A finite oscillator model with equidistant position spectrum based on an extension of

We consider an extension of the real Lie algebra su(2) by introducing a parity operator P and a parameter c. This extended algebra is isomorphic to the Bannai-Ito algebra with two parameters equal to zero. For this algebra we classify all unitary finite-dimensional representations and show their relation with known representations of su(2). Moreover, we present a model for a one-dimensional finite oscillator based on the odd-dimensional representations of this algebra. For this model, the spectrum of the position operator is equidistant and coincides with the spectrum of the known su(2) oscillator. In particular the spectrum is independent of the parameter c while the discrete position wavefunctions, which are given in terms of certain dual Hahn polynomials, do depend on this parameter.

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.

The total angular momentum algebra related to the S3 Dunkl Dirac equation

Abstract We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S 3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A 2 , with corresponding Weyl group S 3 , the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra s o ( 3 ) , incorporating elements of S 3 . This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible representations and determine the conditions for the representations to be unitarizable. The class of unitary irreducible representations admits a natural realization acting on a representation space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.

Unique characterization of the Fourier transform in the framework of representation theory

In this paper we elaborate upon the investigation initiated in [3] of typical and distinctive properties of the Fourier transform (FT), in particular the crucial role played by the Howe dual pair (O(m), sl(2)). We prove in detail a result on the unique characterization of the FT making extensive use of a representation of the Lie algebra sl(2). As an example, we consider the case m = 1. We refer to [3] for a detailed study involving the derivation of a class of operators portraying FT symmetry properties.

The Wigner distribution function for the

Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is defined on discrete phase-space (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the finite oscillator. The first expression for the matrix elements involves sums over squares of Krawtchouk polynomials, and follows from standard techniques. We also manage to present a second solution, where the matrix elements are evaluations of Dyck polynomials. These Dyck polynomials are defined in terms of the well-known Dyck paths. This combinatorial expression of the pre-Wigner matrix elements turns out to be particularly simple.

\mathfrak {su}(2)

In [R. Oste and J. Van der Jeugt, arXiv: 1507.01821 [math-ph]] we classified all pairs of recurrence relations in which two (dual) Hahn polynomials with different parameters appear. Such pairs are referred to as (dual) Hahn doubles, and the same technique was then applied to obtain all Racah doubles. We now consider a special case concerning the doubles related to Racah polynomials. This gives rise to an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. Just as it was the case for (dual) Hahn doubles, the resulting two-diagonal matrix can be used to construct a finite oscillator model. We discuss some properties of this oscillator model, give its (discrete) position wavefunctions explicitly, and illustrate their behavior by means of some plots.

finite oscillator and Dyck paths

In Oste and Van der Jeugt, SIGMA, 12 (2016), [13], we classified all pairs of recurrence relations connecting two sets of Hahn, dual Hahn or Racah polynomials of the same type but with different parameters. We examine the algebraic relations underlying the Racah doubles and find that for a special case of Racah doubles with specific parameters this is given by the so-called Racah algebra.