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 theory to compute the Green functions within a rigorous mathematical theory. We 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 the Green functions as functions of the energy keeps track of the so-called “incoming” and “outgoing” boundary conditions.