The effect of coupling a shadow channel to a real channel is studied in the framework of nonrelativistic potential-scattering theory. For simplicity, $s$-wave scattering with energy-independent $\ensuremath{\delta}$-function potentials is considered. This model has the virtue of being exactly solvable in configuration space. The main features of the solution which are analogous to those expected in a relativistic field theory whose divergences have been removed using Sudarshan's shadow states, are as follows. The coupled problem reduces in the real channel to scattering by an effective energy-dependent pseudopotential. The physical scattering amplitude still satisfies elastic unitarity at all energies, but is piecewise analytic, with a point of nonanalyticity occurring at the threshold for the shadow channel.