Complementarity versus coordinate transformations: mapping between pseudo-Hermiticity and weak pseudo-Hermiticity

英文摘要：

\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 $\widetildeη_+$ (resp. weak pseudo-Hermiticity $\widetildeη_-$). 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 $\widetildeη_+$ and $\widetildeη_-$. A special factorization $η_+=η_-^\dagger η_-$ is discussed in the case of a constant mass and some Bäcklund transformations are derived.