Complex phase space formulation of supersymmetric quantum mechanics: analysis of shape-invariant potentials

With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.