Potentials from the Scattering Amplitude at Fixed Energy: Matrix Methods

Abstract

The simplest way to introduce a function $f_{{V_0}}^V(r,r')$ which is a solution of (XI.3.1) and which depends on a sequence of parameters {c_μ} is to construct a linear combination of products $\varphi _\mu ^{{V_0}}(r)\varphi _\mu ^{{V_0}}(r')$ with coefficients c_μ. An alternative method might introduce products with strong consistency conditions. In all cases, a relation between {c_μ} and {δ_l} must be found by investigating the asymptotic behavior of $\varphi _l^V(r)$. This aim is achieved in the matrix methods we present in this chapter. They yield potentials in special classes that are defined by the nature of {μ}. We strictly limit our study to real potentials.