Anti-PT-symmetric harmonic oscillator and its relation to the inverted harmonic oscillator

TL;DR

This paper explores the quantum dynamics of the inverted harmonic oscillator by connecting it to an anti-PT-symmetric harmonic oscillator, providing a consistent framework for eigenfunctions and classical correspondence.

Contribution

It introduces a Dyson transformation linking the inverted oscillator to an anti-PT-symmetric system, ensuring proper normalization and a complete eigenproblem description.

Findings

Eigenfunctions are normalized via pseudo-scalar product.

Wave packets relate to generalized coherent states.

Mean values follow classical trajectories.

Abstract

We treat the quantum dynamics of a harmonic oscillator as well as its inverted counterpart in the Schr\"odinger picture. Generally in the most papers of the literature, the inverted harmonic oscillator is formally obtained from the harmonic oscillator by the replacement of {\omega} to i{\omega}, this leads to unbounded eigenvectors. This explicitly demonstrates that there are some unclear points involved in redefining the variables in the harmonic oscillator inversion. To remedy this situation, we introduce a scaling operator (Dyson transformation) by connecting the inverted harmonic oscillator to an anti-PT-symmetric harmonic oscillator, we obtain the standard quasi-Hermiticity relation which would ensure the time invariance of the eigenfunction's norm. We give a complete description for the eigenproblem. We show that the wavefunctions for this system are normalized in the sense of the pseudo-scalar product. A Gaussian wave packet of the inverted oscillator is investigated by using the ladder operators method. This wave packet is found to be associated with the generalized coherent state that can be crucially utilized for investigating the mean values of the space and momentum operators. We find that these mean values reproduce the classical motion.