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

Abstract

\noindent We study the concept of the complementarity, introduced by Bagchi and Quesne in [Phys. Lett. A {\bf 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 $V_\pm(x)$ under both concepts of pseudo-Hermiticity $\widetilde\eta_+$ (resp. weak pseudo-Hermiticity $\widetilde\eta_-$). We show that the concept of complementarity can be understood and interpreted as a coordinate transformation through their respective generating functions. As consequence, a similarity transformation which implements coordinate transformations is obtained. We show that the similarity transformation is set up as fundamental relationship connecting both $\widetilde\eta_+$ and $\widetilde\eta_-$. A special factorization $\eta_+=\eta_-^\dagger \eta_-$ is discussed in the case of a constant mass and some B\"acklund transformations are derived.