Rationally-extended Dunkl oscillator on the line

Abstract

It is shown that the extensions of exactly-solvable quantum mechanical problems connected with the replacement of ordinary derivatives by Dunkl ones and with that of classical orthogonal polynomials by exceptional orthogonal ones can be easily combined. For such a purpose, the example of the Dunkl oscillator on the line is considered and three different types of rationally-extended Dunkl oscillators are constructed. The corresponding wavefunctions are expressed in terms of exceptional orthogonal generalized Hermite polynomials, defined in terms of the three different types of Xm-Laguerre exceptional orthogonal polynomials. Furthermore, the extended Dunkl oscillator Hamiltonians are shown to be expressible in terms of some extended Dunkl derivatives and some anharmonic oscillator potentials.

Liénard type nonlinear oscillators and quantum solvability

“…where ¢ = d dx and we seek for the solution for arbitrary values of the ordering parameters ā and b [24]. In contrast to the conventional way of expressing the Hamiltonian (110), researchers have recently used Dunkl formalism [41] in the quantum framework to explore the behavior of nonlinear oscillators [42][43][44][45].…”

“… which has been solved for both positive and negative values of λ. Note that the sign in (67) corresponds to the case λ = -|λ| in equation (44).…”

Construction of angular-dependent potentials from trigonometric Pöschl-Teller systems within the Dunkl formalism

“…By means of this process, we can construct a large variety of relativistic and nonrelativistic quantum systems within the Dunkl context. Particular cases that have recently been studied include the harmonic oscillator in the two-dimensional plane [11,12], the planar Coulomb system [13], the Klein-Gordon equation for several interactions [14], rational extensions of the Dunkl oscillator [15], the relativistic harmonic oscillator [16,17], and three-dimensional systems that allow for separation of variables in the Schrödinger equation [18]. In the present work, we focus on the latter type of systems.…”