Complementarity vs coordinate transformations: Mapping between pseudo-Hermiticity and weak pseudo-Hermiticity
Journal of Mathematical Physics , Volume 62; doi:10.1063/5.0036401
Abstract: 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 η̃+ (respectively, weak pseudo-Hermiticity η̃−). 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 η̃+ and η̃−. A special factorization η+=η−†η− is discussed in the constant mass case, and some Bäcklund transformations are derived.
Keywords: coordinate transformations / functions / complementarity / weak pseudo Hermiticity
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