On quantum scattering by δ′(x) and quasi ′(x) distributions

Abstract

The zero-range interaction U(x) occurring in the one-dimensional, time-independent Schrödinger equation is regarded as a smoothed distribution characterized by a tiny length scale b such that the origin becomes an ordinary point. A neighbourhood around the origin is scanned by defining inner demarcation points a± = ±b/N and outer demarcation points b± ≡ ±Nb with N ≫ 1. Then a sequence of simple Lemmas permits (i) construction of a systematic procedure for simultaneously solving the scattering wave function ψ(0) at the origin, its derivative ψ(0) there, the transmission amplitude B, as well as the reflection amplitude D; and (ii) unambiguous application to scattering by the previously known δ′(x) and newly proposed quasi δ′(x) potentials in the Cauchy representation of various distributions. © 2007 NRC Canada.