We look for the solution to the eigenvalue problem, in a recently introduced phase-space representation, for the quantum particle in a linear potential. We find that the solution is not unique and that some of these functions correspond to the eigenfunctions in coordinate space and that others correspond to the classical limit.

Using a special phase space representation, we propose a factorization of the Hamiltonian in this representation in spite of its more complex structure. Also, based on the link existing between the intertwining technique and SUSY we find the phase space supercharges and the supersymmetric partner Hamiltonian.

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