Rigorous treatment of scattering by quantum zero-range interactions

Abstract

A new rigorous treatment of solving the Schrödinger scattering equation for a general contact potential U(x) has been developed. It is proved that if U(x) is regarded as a distribution characterized by a tiny length scale b then the wave function ψ and its derivative ψ′ become continuous across the apparent singular point x = 0. Hence, the reflection and transmission coefficients can be constructed for many cases namely usual δ(x), controversial δ′(x) and entirely new quasi δ′(x) interactions.