Effect of the two-parameter generalized Dunkl derivative on the two-dimensional Schrödinger equation

Abstract

In this paper, we introduce a generalization of the Dunkl derivative with two parameters to study the Schrödinger equation in Cartesian and polar coordinates in two dimensions. The eigenfunctions and the energy spectrum for the harmonic oscillator and the Coulomb problem are derived in an analytical way and it is shown that our results are properly reduced to those previously reported for the Dunkl derivative with a single parameter.

Approximate bound state solutions of the Dunkl-Schrödinger equation for a hyperbolic double-well interaction

“…Within this deformation, the Hamiltonian, and the associated Schrödinger equation can be constructed, as well as a weighted Hilbert space containing the wavefunctions, see [22] for the one-dimensional scenario. A vast amount of quantum systems has been studied within the Dunkl formalism, including the harmonic oscillator [23,24], a two-dimensional system [25], the massless Dirac equation [26], the Dunkl-Dirac oscillator [27], just to name a few. In addition, the path integral formalism was developed for the Dunkl context [28], and rational extensions were constructed [29,30].…”

The Condensation of Ideal Bose Gas in a Gravitational Field in the Framework of Dunkl-Statistic

“…The dynamics of electrons in a graphene layer which is under the effect of external magnetic field is derived in [55]. There are many more studies in the literature that takes into account the Dunkl formalism as it allows itself to discuss parity-dependent solutions simultaneously [56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73].…”

Quasi-exactly solvable potentials in Wigner-Dunkl quantum mechanics

“…Coherent states [12], a generalization of shape invariance in supersymmetric quantum mechanics [13], and relativistic effects [14,15] have also been considered. The possibility of using Dunkl derivatives with two or three parameters has been investigated [16,17]. Very recently, some rationally extended potentials, whose wave functions can be expressed in terms of exceptional orthogonal polynomials have been studied in the Wigner-Dunkl formalism [18,19].…”