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We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions,defined as a $\lambda-$deformation of the N-dimensional Dunkl oscillator. Thisdeformation can be interpreted either as the introduction of a non-constantcurvature related to $\lambda$ on the underlying space or, equivalently, as aDunkl oscillator with a position-dependent mass function. This new quantummodel is shown to be exactly solvable in arbitrary dimension N, and itseigenvalues and eigenfunctions are explicitly presented. Moreover, it is shownthat in the two-dimensional case both the Darboux III and the Dunkl oscillatorscan be separately coupled with a constant magnetic field, thus giving rise totwo new exactly solvable quantum systems in which the effect of aposition-dependent mass and the Dunkl derivatives on the structure of theLandau levels can be explicitly studied. Finally, the whole 2D Dunkl-DarbouxIII oscillator is coupled with the magnetic field and shown to define anexactly solvable Hamiltonian, where the interplay between the$\lambda$-deformation and the magnetic field is explicitly illustrated.

The Dunkl oscillator on a space of nonconstant curvature: an exactly solvable quantum model with reflections