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.

中文翻译：

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互补与坐标变换：伪厄米性和弱伪厄米性之间的映射

我们在本文中研究了由 Bagchi 和 Quesne [Phys. 莱特。A 301 , 173 (2002)]，当系统具有与位置相关的质量时，在坐标变换的严格数学观点中的伪厄米性和弱伪厄米性之间。我们首先确定，在修正动量下，在两个伪厄米性概念下识别复势V _± ( x )的生成函数${\tilde{\eta }}_{+}$ （分别为弱伪厄米性 ${\tilde{\eta }}_{-}$）。我们表明互补性的概念可以通过它们各自的生成函数来理解和解释为坐标变换。结果，获得了实现坐标变换的相似变换。我们表明，相似变换被设置为连接两者的基本关系${\tilde{\eta }}_{+}$ 和 ${\tilde{\eta }}_{-}$. 特殊分解${\eta }_{+}={\eta }_{-}^{†}{\eta }_{-}$ 在恒质量情况下讨论，并导出了一些 Bäcklund 变换。