The Three-Dimensional Inverse Problem

Abstract

In a time-independent formulation of nonrelativistic scattering, one starts with the Schrödinger equation

$$ - \Delta \psi + V(r)\psi = E\psi $$

(XIV.1.1)

for the wave function Ψ, and one assumes that the properties of V(r) are sufficient to guarantee the following asymptotic form of Ψ:

$$\psi \left( {k,r} \right) = \exp \left[ {ik \cdot r} \right] + {r^{ - 1}}\exp \left[ {ikr} \right]A\left( {\hat r,k} \right) + o\left( {{r^{ - 1}}} \right).$$

(XIV.1.2)

(For any vector y, we write y for its length, P for v/u. In the scattering amplitude, we use k’ for kr, and, more generally, we use k’ for a vector of length k, and which is not k).