On the spectrum of the Dirac operator and the existence of discrete eigenvalues for the defocusing nonlinear Schrödinger equation

We revisit the scattering problem for the defocusing nonlinear Schrodinger equation with constant, nonzero boundary conditions at infinity, i.e., the eigenvalue problem for the Dirac operator with nonzero rest mass. By considering a specific kind of piecewise constant potentials we address and clarify two issues, concerning: (i) the (non)existence of an area theorem relating the presence/absence of discrete eigenvalues to an appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase difference of the potential from the continuous spectrum.