Conserved correlation in image-symmetric systems: Scattering and bound states Original Research Article

For one-dimensional image-symmetric systems, it is observed that the non-local product image, 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 image-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 image CPA laser (Longhi, 2010) . The interpretation of ‘left’ and ‘right’ states leads to a Hermitian image-matrix, resulting in the non-conservation of the ‘flux’. This further satisfies a ‘duality’ condition, identical to the optical analogues (Paasschens et al., 1996) . 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 image-symmetry is preserved, it leads to stationary states with real energy, naturally interpretable as bound states. The broken image-symmetric phase is also captured by this correlation, with complex-conjugate pair of energies, interpreted as resonances.