Complementarity versus 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 ${\tilde{\eta }}_{+}$ (resp. 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 consequence, a similarity transformation which implements coordinate transformations is obtained. We show that the similarity transformation is set up as fundamental relationship connecting both ${\tilde{\eta }}_{+}$ and ${\tilde{\eta }}_{-}$. A special factorization ${\eta }_{+}={\eta }_{-}^{†}{\eta }_{-}$ is discussed in the case of a constant mass and some Bäcklund transformations are derived.

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