Complementarity vs coordinate transformations: Mapping between pseudo-Hermiticity and weak pseudo-Hermiticity

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.

Higher-order phase-space moments for off-diagonal rotating Morse oscillators

“…Our results in this paper, as in the previous work [1], were limited to a Hermitian Morse oscillator with a constant-mass. Therefore, a natural extension of these works would be to consider quantum systems with position-dependent mass (PDM) [28], subjected to either PT-symmetric or pseudo-hermiticity [31][32][33], to examine how the PDM-function M(x) and the discrete symmetry pseudo hermiticity  affect moment measurements. Some of these ideas are currently in progress.…”