Classical Trajectories for Complex Hamiltonians

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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)ε (ε≥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 ε and on the initial conditions. A system for classifying complex orbits is presented.