We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modeling the propagation of electromagnetic waves traveling in a waveguide half and half filled with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter zeta, we show that the equivalent Hermitian Hamiltonian has the form p(2)/2m+(zeta(2)/2)Sigma(infinity)(n=0){alpha(n)(x),p(2n)} with alpha(n)(x) vanishing outside an interval that is three times larger than the support of v(x), i.e., in 2/3 of the physical interaction region the potential v(x) vanishes identically. We provide a physical interpretation for this unusual behavior and comment on the classical limit of the system.
Mostafazadeh, A. "Application of Pseudo-Hermitian Quantum Mechanics to a Pt-Symmetric Hamiltonian with a Continuum of Scattering States." Journal of Mathematical Physics 46.10 (2005): 15.