Classical Trajectories for Complex Hamiltonians

TL;DR

This paper investigates the classical trajectories of complex, non-Hermitian Hamiltonians with unbroken $ ext{PT}$ symmetry, revealing intricate structures dependent on parameters and initial conditions, and introduces a classification system for these orbits.

Contribution

It provides a detailed analysis of classical trajectories for a class of complex Hamiltonians, highlighting their complex structures and proposing a classification method.

Findings

Classical trajectories exhibit rich, elaborate structures.

Trajectory behavior depends sensitively on parameter $psilon$ and initial conditions.

A classification system for complex orbits is developed.

Abstract

It has been found that complex non-Hermitian quantum-mechanical Hamiltonians may have entirely real spectra and generate unitary time evolution if they possess an unbroken $\cP\cT$ symmetry. A well-studied class of such Hamiltonians is $H= p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$). This paper examines the underlying classical theory. Specifically, it explores the possible trajectories of a classical particle that is governed by this class of Hamiltonians. These trajectories exhibit an extraordinarily rich and elaborate structure that depends sensitively on the value of the parameter $\epsilon$ and on the initial conditions. A system for classifying complex orbits is presented.