Formal and precise derivation of the Green functions for a simple potential

In formal scattering theory, the Green functions are obtained as solutions of a distributional equation. In this paper, we use the Sturm-Liouville the ory to compute the Green functions within a rigorous mathematical theory. W e shall show that both the Sturm-Liouville theory and the formal treatment yield the same Green functions. We shall also show how the analyticity of t he Green functions as functions of the energy keeps track of the so-called "incoming" and "outgoing" boundary conditions. (C) 2001 Elsevier Science Lt d. All rights reserved.