Complementarity vs coordinate transformations: Mapping between pseudo-Hermiticity and weak pseudo-Hermiticity

Abstract

We investigate in this paper the concept of complementarity, introduced by Bagchi and Quesne [Phys. Lett. A 301, 173 (2002)], between pseudo-Hermiticity and weak pseudo-Hermiticity in a rigorous mathematical viewpoint of coordinate transformations when a system has a position-dependent mass. We first determine, under the modified-momentum, the generating functions identifying the complexified potentials V_±(x) under both concepts of pseudo-Hermiticity ${\tilde{\eta }}_{+}$ (respectively, weak pseudo-Hermiticity ${\tilde{\eta }}_{-}$ ). We show that the concept of complementarity can be understood and interpreted as a coordinate transformation through their respective generating functions. As a consequence, a similarity transformation that implements coordinate transformations is obtained. We show that the similarity transformation is set up as a fundamental relationship connecting both ${\tilde{\eta }}_{+}$ and ${\tilde{\eta }}_{-}$ . A special factorization ${\eta }_{+}={\eta }_{-}^{†}{\eta }_{-}$ is discussed in the constant mass case, and some Bäcklund transformations are derived.