Schrodinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

Schrodinger equations with non-Hermitian, but PT -symmetric quantum potentials V(x) found, recently, a new field of applicability in classical optics. The potential acquired there a new physical role of an anomalous refraction index. This turned attention to the nonlinear Schrodinger equations in which the interaction term becomes state-dependent, V(x) -> W(psi(x),x). Here, the state-dependence in W(psi(x),x) is assumed logarithmic, and some of the necessary mathematical assumptions, as well as some of the potential phenomenological consequences of this choice are described. Firstly, an elementary single-channel version of the nonlinear logarithmic model is outlined in which the complex self-interaction W(psi(x),x) is regularized via a deformation of the real line of x into a self-consistently constructed complex contour C. The new role played by PT -symmetry is revealed. Secondly, the regularization is sought for a multiplet of equations, coupled via the same nonlinear self-interaction coupling of channels. The resulting mathematical structures are shown to extend the existing range of physics covered by the logarithmic Schrodinger equations.