Effects of complex time on periodic and nonperiodic classical trajectories of one-dimensional Hamiltonian systems

Abstract

In recent years, much research has been carried out on extending both quantum mechanics and classical mechanics into the complex domain by making parameters of real hermitian Hamiltonians or total energy of the system complex. In this paper we investigate the effects of complex time on periodic and nonperiodic trajectories of both hermitian and nonhermitian one-dimensional classical Hamiltonian systems. Most of the periodic classical trajectories of real hermitian systems turn into nonperiodic and open when the energy or the parameters of the potential become complex. We show that when time is taken as a complex quantity with a specific fixed phase angle or as a specific complex function, nonperiodic trajectories become periodic and closed. Furthermore, we show that real hermitian systems, such as H = p2/2m + x4 + bx3 + cx2 + dx (b, c, and d are real quantities) possess classical periodic trajectories for real energies even when time is complex (i.e., t = treiτ). It was found that there is a discrete set of τ values for which the trajectories of the preceding system are closed and periodic and periods associated with them form a countably infinite set.