Effects of complex parameters on classical trajectories of Hamiltonian systems

TL;DR

This paper investigates how complex parameters influence the periodicity of classical trajectories in Hamiltonian systems, revealing discrete conditions for closed trajectories and energy quantization in complex quartic Hamiltonians.

Contribution

It demonstrates that classical trajectories are closed only on discrete parameter curves and energies in complex quartic Hamiltonians, extending previous results to more general complex systems.

Findings

Classical trajectories are closed only on discrete parameter curves in the complex b-plane.

Classical energy becomes discretized for given complex parameters due to periodicity conditions.

Trajectories are periodic for specific angles in complex Hamiltonians with real energies.

Abstract

Anderson $\textit{et al}$ have shown that for complex energies, the classical trajectories of $\textit{real}$ quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that, when time is complex $t$ $(t=t_{r}e^{i\theta _{\tau }}),$ certain real hermitian systems possess close periodic trajectories only for a discrete set of values of $\theta _{\tau }$. On the other hand it is generally true that even for real energies, classical trajectories of non $\mathcal{PT}$- symmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper we show that for given real energy, the classical trajectories of $\textit{complex}$ quartic Hamiltonians $H=p^{2}+ax^{4}+bx^{k}$, (where $a$ is real, $b$ is complex and $k=1$ $or$ $2$) are closed and periodic only for a discrete set of parameter curves in the complex $b$-plane. It was further found that given complex parameter $b$, the classical trajectories are periodic for a discrete set of real energies (i.e. classical energy get discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of $\textit{complex}$ Hamiltonian $H=p^{2}+\mu x^{4}, (\mu=\mu _{r}e^{i\theta })$ are periodic when $\theta =4 tan^{-1}[(n/(2m+n))]$ for $\forall $ $ n$ and $m\in \mathbb{Z}$.