Effects of complex parameters on classical trajectories of Hamiltonian systems

Abstract

Anderson $\text{et al}$ have shown that for complex energies, the classical trajectories of $\text{real}$ 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 $\mathcal{P}\mathcal{T}$- 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 $\text{complex}$ quartic Hamiltonians $H=p^2+a{x}^4+b{x}^k$, (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 $\text{complex}$ Hamiltonian $H=p^2+\mu x^4,(\mu ={\mu }_r{e}^{i\theta })$ are periodic when $\theta =4ta{n}^{-1}[(n/(2m+n))]$ for $\forall$ $n$ and $m\in \mathbb{Z}$.

arXiv comment9 pages, 2 tables, 6 figures

References in corpus (20)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• PT-Symmetric Quantum Mechanics
• Introduction to PT-Symmetric Quantum Theory
• Supersymmetry and the spontaneous breakdown of PT symmetry
• Scalar Quantum Field Theory with Cubic Interaction
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• PT-Symmetric Extension of the Korteweg-de Vries Equation
• Classical Trajectories for Complex Hamiltonians
• Complexified Dynamical Systems
• Complex Trajectories of a Simple Pendulum
• Quantum effects in classical systems having complex energy
• Spontaneous Breaking of Classical PT Symmetry
• On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex $PT$-Symmetric Quartic Hamiltonian
• Chaotic systems in complex phase space
• Periodic orbits for classical particles having complex energy
• Study of classical mechanical systems with complex potentials