24 research outputs found
A finite oscillator model with equidistant position spectrum based on an extension of su(2)
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
Motzkin paths, Motzkin polynomials and recurrence relations
We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce sonic properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof
Tridiagonal test matrices for eigenvalue computations : two-parameter extensions of the Clement matrix
The Clement or Sylvester-Kac 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.Comment: This is a preprint of a paper whose final and definite form is in
Journal of Computational and Applied Mathematic
A finite quantum oscillator model related to special sets of Racah polynomials
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
Laplace and Dirac operators, symmetry algebras, and their use in Fourier transforms and quantum oscillator models
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 E.I., Stoilova N.I., 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 Christoffel 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
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, which are 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 anti-commute 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.Comment: 39 pages, final versio
The total angular momentum algebra related to the Dunkl Dirac equation
We consider the symmetry algebra generated by the total angular momentum
operators, appearing as constants of motion of the 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 , with corresponding
Weyl group , 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 , incorporating
elements of . 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-Kowalevsky extension theorem we obtain explicit expressions for these
eigenfunctions in terms of Jacobi polynomials.Comment: 29 pages, 1 figure; New title, introduction and physics contex
Exactly solvable model of the one-dimensional confined harmonic oscillator
An exactly solvable model of the one-dimensional quantum harmonic oscillator confined in a box with infinite walls is constructed. We have found explicit expressions of the non-equidistant energy spectrum as well as stationary states wavefunctions in both momentum and position configuration spaces. It is shown that they are expressed through continuous q-Hermite polynomials. We have also found an explicit expression for the kernel of the finite-continuous Fourier transform between these two representation spaces
The Wigner distribution function for the su(2) finite oscillator and Dyck paths
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 su(2) 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