Classical Trajectories for Complex Hamiltonians

Details Classical Trajectories for Complex Hamiltonians Classical Trajectories for Complex Hamiltonians

2006-02-14

arxiv.org/abs/math-ph/0602040v1

J. Phys. A: Math. Gen. 39, 4219-4238 (2006)

Physics

Mathematical Physics

24 pages, 34 figures

Scientific paper

10.1088/0305-4470/39/16/009

It has been found that complex non-Hermitian quantum-mechanical Hamiltonians may have entirely real spectra and generate unitary time evolution if they possess an unbroken $\cP\cT$ symmetry. A well-studied class of such Hamiltonians is $H= p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$). This paper examines the underlying classical theory. Specifically, it explores the possible trajectories of a classical particle that is governed by this class of Hamiltonians. These trajectories exhibit an extraordinarily rich and elaborate structure that depends sensitively on the value of the parameter $\epsilon$ and on the initial conditions. A system for classifying complex orbits is presented.

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