Quasi-exactly solvable potentials in Wigner-Dunkl quantum mechanics

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 $n+1$ known eigenstates for any $n\in \N$. 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 $n+1$ known eigenstates are obtained, whereas, in the latter, sets of $n+1$ potentials associated with a given energy are derived.