θ(x,p)-deformation of the harmonic oscillator in a 2D-phase space
Abstract
This work addresses a θ(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 q-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 q- Hermite polynomials. Relevant matrix elements are computed. Finally, a su(2)-algebra representation of the considered deformation is investigated and discussed.
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