A PT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{PT}$$\end{document}-symmetric simple harmonic oscillator

We consider a simple harmonic oscillator with non-Hermitian term and study it classically and quantum mechanically. We conclude that this version of oscillator, which breaks parity and time reversal, displays all the features possessed by the usual harmonic oscillator. In particular, we calculate its spectrum, adiabatic invariance and Wigner functions, and show that there is a consistency between the classical and quantum descriptions.