Classical trajectories for complex Hamiltonians

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 symmetry. A well-studied class of such Hamiltonians is H = p2 + x2(ix) ( ≥ 0). 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 and on the initial conditions. A system for classifying complex orbits is presented.