Understanding the Complex Position in a PT-symmetric Oscillator

TL;DR

This paper investigates the complex coordinates in PT-symmetric oscillators, revealing how real and imaginary parts relate to observations and non-Hermiticity, and introduces a new complex P-transformation.

Contribution

It presents a novel analysis of complex positions in PT-symmetric systems and proposes an extended P-transformation for better understanding.

Findings

Complex position information is fully accessible in PT-symmetric oscillators.

Real part of position is observable; imaginary part relates to non-Hermiticity.

Introduces a new complex P-transformation extending traditional symmetry concepts.

Abstract

We study how to understand the complex coordinates involved in the non-Hermitian but PT-symmetric systems. We explore a PT-symmetric oscillator model to show that the entire information on the complex position is attainable. Its real part is from the observation while its imaginary part is from the non-Hermiticity parameter. We also propose a new complex extension of P-transformation and T-transformation (the `parity' and `time reflection' respectively). Particularly, the P-transformation realizes the left-right reflection in the complex plane.