Potentials from the Scattering Amplitude at Fixed Energy: Matrix Methods

Abstract

The simplest way to introduce a function

$$f_{{v_0}}^v\left( {r,r'} \right)$$

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

$$\varphi _\mu ^{{V_0}}\left( r \right)\varphi _\mu ^{{V_0}}\left( {r'} \right)$$

with coefficients c _µ . An alternative method might introduce products

$$\varphi _{ - \mu }^{{V_0}}\varphi _\mu ^{{V_0}},$$

with strong consistency conditions. In all cases, a relation between {c _µ } and {δ _l } must be found by investigating the asymptotic behavior of

$$\varphi _l^V\left( r \right)$$

. 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.