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

TL;DR

This paper explores the mathematical relationship between pseudo-Hermiticity and weak pseudo-Hermiticity, showing they are connected via coordinate transformations, with implications for systems with position-dependent mass and similarity transformations.

Contribution

It provides a rigorous mathematical framework linking pseudo-Hermiticity and weak pseudo-Hermiticity through coordinate transformations and generating functions.

Findings

Complementarity interpreted as coordinate transformation.

Derived similarity transformation connecting both concepts.

Discussed factorization and Bäcklund transformations in specific cases.

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.