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https://www.ias.ac.in/article/fulltext/pram/082/06/0973-0983

Classical trajectories; non-Hermitian; periodic trajectories.

Anderson et al have shown that for complex energies, the classical trajectories of 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-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 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 gets 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 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 $m \epsilon \mathbb{Z}$.

Asiri Nanayakkara¹ Thilagarajah Mathanaranjan^{1 2}

1. Institute of Fundamental Studies, Hanthana Road, Kandy, Sri Lanka
2. Department of Mathematics, University of Jaffna, Jaffna, Sri Lanka

Published

June 2014

Manuscript received

19 September 2013

Manuscript revised

28 October 2013

Accepted

25 November 2013

Early published

28 May 2014