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

References in corpus (5)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• PT-Symmetric Quantum Mechanics
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Supersymmetry and the spontaneous breakdown of PT symmetry

Cited by in corpus (28)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• PT-Symmetric Extension of the Korteweg-de Vries Equation
• Complex Correspondence Principle
• Complexified Dynamical Systems
• Complex Trajectories of a Simple Pendulum
• Quantum effects in classical systems having complex energy
• Spontaneous Breaking of Classical PT Symmetry
• Chaotic systems in complex phase space
• Real Description of Classical Hamiltonian Dynamics Generated by a Complex Potential
• Quantum tunneling as a classical anomaly
• $J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach
• Probability Density in the Complex Plane
• Dynamical systems on infinitely sheeted Riemann surfaces
• Exceptional points in quantum and classical dynamics
• Periodic orbits for classical particles having complex energy
• Classical Particle in a Complex Elliptic Potential
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• Conduction bands in classical periodic potentials
• Does the complex deformation of the Riemann equation exhibit shocks?
• Classical oscillator with position-dependent mass in a complex domain
• On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)
• Study of classical mechanical systems with complex potentials
• Coherent States and Modified de Broglie-Bohm Complex Quantum Trajectories
• Nonlinear Integral-Equation Formulation of Orthogonal Polynomials
• Quantum counterpart of spontaneously broken classical PT symmetry
• Instanton calculus without equations of motion: semiclassics from monodromies of a Riemann surface
• Effects of complex parameters on classical trajectories of Hamiltonian systems