9 research outputs found
From slq(2) to a Parabosonic Hopf Algebra
A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl₋₁(2), this algebra encompasses the Lie superalgebra osp(1|2). It is obtained as a q=−1 limit of the slq(2) algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible representations. It possesses a noncocommutative coproduct. The Clebsch-Gordan coefficients (CGC) of sl₋₁(2) are obtained and expressed in terms of the dual −1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization
Supersymmetric Quantum Mechanics with Reflections
We consider a realization of supersymmetric quantum mechanics where
supercharges are differential-difference operators with reflections. A
supersymmetric system with an extended Scarf I potential is presented and
analyzed. Its eigenfunctions are given in terms of little -1 Jacobi polynomials
which obey an eigenvalue equation of Dunkl type and arise as a q-> -1 limit of
the little q-Jacobi polynomials. Intertwining operators connecting the wave
functions of extended Scarf I potentials with different parameters are
presented.Comment: 17 page
An Algebraic Model for the Multiple Meixner Polynomials of the First Kind
An interpretation of the multiple Meixner polynomials of the first kind is
provided through an infinite Lie algebra realized in terms of the creation and
annihilation operators of a set of independent oscillators. The model is used
to derive properties of these orthogonal polynomials
Jordan algebras and orthogonal polynomials
We illustrate how Jordan algebras can provide a framework for the
interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi
polynomials are eigenfunctions of a first order operator of Dunkl type. We
consider an algebra that has this operator (up to constants) as one of its
three generators and whose defining relations are given in terms of
anticommutators. It is a special case of the Askey-Wilson algebra AW(3). We
show how the structure and recurrence relations of the big -1 Jacobi
polynomials are obtained from the representations of this algebra. We also
present ladder operators for these polynomials and point out that the big -1
Jacobi polynomials satisfy the Hahn property with respect to a generalized
Dunkl operator.Comment: 11 pages, 30 reference
An infinite family of superintegrable Hamiltonians with reflection in the plane
We introduce a new infinite class of superintegrable quantum systems in the
plane. Their Hamiltonians involve reflection operators. The associated
Schr\"odinger equations admit separation of variables in polar coordinates and
are exactly solvable. The angular part of the wave function is expressed in
terms of little -1 Jacobi polynomials. The spectra exhibit "accidental"
degeneracies. The superintegrability of the model is proved using the
recurrence relation approach. The (higher-order) constants of motion are
constructed and the structure equations of the symmetry algebra obtained.Comment: 19 page
Discrete series representations for sl(2|1), Meixner polynomials and oscillator models
We explore a model for the one-dimensional quantum oscillator based upon the
Lie superalgebra sl(2|1). For this purpose, a class of discrete series
representations of sl(2|1) is constructed, each representation characterized by
a real number beta>0. In this model, the position and momentum operators of the
oscillator are odd elements of sl(2|1) and their expressions involve an
arbitrary parameter gamma. In each representation, the spectrum of the
Hamiltonian is the same as that of the canonical oscillator. The spectrum of
the momentum operator can be continuous or infinite discrete, depending on the
value of gamma. We determine the position wavefunctions both in the continuous
and discrete case, and discuss their properties. In the discrete case, these
wavefunctions are given in terms of Meixner polynomials. From the embedding
osp(1|2)\subset sl(2|1), it can be seen why the case gamma=1 corresponds to the
paraboson oscillator. Consequently, taking the values (beta,gamma)=(1/2,1) in
the sl(2|1) model yields the canonical oscillator.Comment: (some minor misprints were corrected in this version