Remarks on boundary conditions for scalar scattering

Abstract

The question is discussed whether potential scattering problems can be treated as boundary value problems associated with differential equations, as is sometimes suggested in the literature. We show that, except in some very special cases, this is not possible. The values of the wave function and its normal derivative on the boundary of a finite-range potential cannot be prescribed arbitrarily but are implicit in the integral equation of potential scattering. We derive two coupled singular integral equations for the boundary values for the case when the scattering potential is homogeneous.