Quasi-exactly solvable potentials in Wigner-Dunkl quantum mechanics

Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles - Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Received: 10 January 2024
Accepted: 14 February 2024

Abstract

It is shown that the Dunkl harmonic oscillator on the line can be generalized to a quasi-exactly solvable one, which is an anharmonic oscillator with known eigenstates for any . It is also proved that the Hamiltonian of the latter can also be rewritten in a simpler way in terms of an extended Dunkl derivative. Furthermore, the Dunkl isotropic oscillator and Dunkl Coulomb potentials in the plane are generalized to quasi-exactly solvable ones. In the former case, potentials with known eigenstates are obtained, whereas, in the latter, sets of potentials associated with a given energy are derived.