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 $X_m$-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.

arXiv comment17 pages, no figure; typos corrected, published version

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Cited by in corpus (8)

• The Dunkl-Fokker-Planck Equation in $1+1$ Dimensions
• Time-Dependent Dunkl-Schrödinger Equation with an Angular-Dependent Potential
• A Path Integral Treatment of Time-dependent Dunkl Quantum Mechanics
• One-dimensional Dunkl Quantum Mechanics: A Path Integral Approach
• Dunkl-Klein-Gordon Equation in Higher Dimensions
• Liénard Type Nonlinear Oscillators and Quantum Solvability
• Exact Solutions of the Schrödinger-Dunkl Equation for a Free Particle in a Finite and Infinite Cylindrical Well
• A Generalization of the Parametric Amplifier with Dunkl Derivative: Spectral and Statistical Properties