Classical trajectories for complex Hamiltonians

Abstract

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 PT symmetry. A well-studied class of such Hamiltonians is H = p 2 + x 2 (ix) ǫ (ǫ ≥ 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 ǫ and on the initial conditions. A system for classifying complex orbits is presented.