Classical Trajectories for Complex Hamiltonians

Abstract

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 $H=p^2+x^2(ix{)}^{\epsilon }$ ($\epsilon \ge 0$). 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.

24 pages, 34 figures

Cited by in corpus (8)

• Making Sense of Non-Hermitian Hamiltonians
• Complexified Dynamical Systems
• Spontaneous Breaking of Classical PT Symmetry
• Real Description of Classical Hamiltonian Dynamics Generated by a Complex Potential
• Exceptional points in quantum and classical dynamics
• Classical oscillator with position-dependent mass in a complex domain
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• Nonlinear Integral-Equation Formulation of Orthogonal Polynomials