Coordinate, Momentum and Dispersion operators in Phase space representation

The aim of this paper is to present a study on the representations of coordinate, momentum and dispersion operators in the framework of a phase space representation of quantum mechanics that we have introduced and studied in previous works. We begin in the introduction section with a recall about the concept of representation of operators on wave function spaces. Then, we show that in the case of the phase space representation the coordinate and momentum operators can be represented either with differential operators or with matrices. The explicit expressions of both the differential operators and matrices representations are established. Multidimensional generalization of the obtained results are performed and phase space representation of dispersion operators are given.