Effects of complex parameters on classical trajectories of Hamiltonian systems

Pramana

Asiri Nanayakkara1¹  Thilagarajah Mathanaranjan1¹ 

[1] Institute of Fundamental Studies, Hanthana Road, Kandy, Sri Lanka$$

关键词: Classical trajectories;    non-Hermitian;    periodic trajectories.;   

DOI  :  

学科分类：物理（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $t(t = t_r e^{i𝜃_𝜏})$, certain real Hermitian systems possess close periodic trajectories only for a discrete set of values of 𝜃𝜏. 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 $H = p^2 + 𝜇 x^4$, ($𝜇= 𝜇_r$ e$^{i𝜃}$) are periodic when $𝜃 = 4 an^{−1}$[($n/(2m + n)$)] for $forall n$ and $m 𝜀 mathbb{Z}$.

【 授权许可】

Unknown