Periodic orbits for classical particles having complex energy

TL;DR

This paper demonstrates that in complex classical mechanics, there exists a discrete set of complex energies where particle trajectories are closed and periodic, extending previous understanding of real-energy cases.

Contribution

It identifies and characterizes eigencurves in the complex-energy plane where classical particles exhibit closed, periodic trajectories, revealing new structure in complex classical mechanics.

Findings

Discrete eigencurves correspond to closed, periodic trajectories

Trajectories are open for generic complex energies

Extension of classical mechanics to complex energies

Abstract

This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is complex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic.