\theta(x,p)-deformation of the Harmonic Oscillator in a 2D-phase Space

Abstract

This work addresses a theta(x,p)-deformation of the harmonic oscillator in a 2D-phase space. Specifically, it concerns a quantum mechanics of the harmonic oscillator based on a noncanonical commutation relation depending on the phase space coordinates. A reformulation of this deformation is considered in terms of a ????-deformation allowing to easily deduce the energy spectrum of the induced deformed harmonic oscillator. Then, it is proved that the deformed position and momentum operators admit a one-parameter family of self-adjoint extensions. These operators engender new families of deformed Hermite polynomials generalizing usual ????-Hermite polynomials. Relevant matrix elements are computed. Finally, a ????????(2)-algebra representation of the considered deformation is investigated and discussed.