PT-symmetric cubic anharmonic oscillator as a physical model

Abstract

We perform a perturbative calculation of the physical observables, in particular, pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator and the classical Hamiltonian for the PT-symmetric cubic anharmonic oscillator, H = -1/2m p(2) + 1/2 mu(2) x(2) + i epsilon x(3). Ignoring terms of order epsilon(4) and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass, and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discuss the quantum-classical correspondence for this PT-symmetric system, compute the associated conserved probability density and comment on the issue of factor ordering in the pseudo-Hermitian canonical quantization of the underlying classical system.