Conserved correlation in PTPT-symmetric systems: Scattering and bound states

For one-dimensional PTPT-symmetric systems, it is observed that the non-local   product ψ∗(−x,t)ψ(x,t)ψ∗(−x,t)ψ(x,t), obtained from the continuity equation can be interpreted as a conserved correlation function. This leads to physical conclusions regarding both discrete and continuum states of such systems. Asymptotic states are shown to have necessarily broken PTPT-symmetry, leading to modified scattering and transfer matrices. This yields restricted boundary conditions, e.g.  , incidence from both sides, analogous to that of the proposed PTPT CPA laser (Longhi, 2010) [4]. The interpretation of ‘left’ and ‘right’ states leads to a Hermitian  SS-matrix, resulting in the non-conservation of the ‘flux’. This further satisfies a ‘duality’ condition, identical to the optical analogues (Paasschens et al., 1996) [17]. However, the non-local conserved scalar implements alternate boundary conditions in terms of ‘in’ and ‘out’ states, leading to the pseudo-Hermiticity condition in terms of the scattering matrix. Interestingly, when PTPT-symmetry is preserved, it leads to stationary states with real energy, naturally interpretable as bound states. The broken PTPT-symmetric phase is also captured by this correlation, with complex-conjugate pair of energies, interpreted as resonances.

► Non-local conserved scalar for PT-symmetric systems.
► Restricted asymptotic boundary conditions.
► Conditions for bound, resonance and scattering states.
► Suggestions towards PT-symmetric Norm.
► An example and few suggestions.