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

Research paper by **Vincent X. Genest, Luc Vinet, Alexei Zhedanov**

Indexed on: **25 Jul '13**Published on: **25 Jul '13**Published in: **Mathematical Physics**

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#### Abstract

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.