Non-unitarity and non-reciprocity in scattering from real potentials in presence of confined non-linearity

Investigations of scattering in presence of non-linearity which have just begun require the confinement of both the potential, V(x), and the non-linearity, γf(|ψ|). There could be two options for the confinement. One is the finite support on x in [-L,L] and the other one is on x in [0,L]. Here, we consider real Hermitian potentials and report a surprising disparate behaviour of these two types of confinements. We prove that in the first option the symmetric potential enjoys reciprocity of both reflectivity (R) and transmitivity (T) and their unitarity. More interestingly, the asymmetry in V(x) causes non-unitarity $ R+Tne 1$ and the non-reciprocity (reciprocity) of T (R). On the other hand, the second option of confinement gives rise to an essential non-unitarity even when V(x) is symmetric about a point in [0,L]. In the absence of symmetry there occurs non-reciprocity of both R and T.