Propagator for narrow potential barriers

The Schrödinger equation in integral form is applied to the one-dimensional scattering problem in the case of a narrow potential barrier. Since the kernel can be considered, in a first approximation, separable, an explicit expression for the propagator is found by means of the complementary error function. The problem of a particle confined in a half-space and interacting with a narrow potential barrier is also considered and solved in an approximate way.