Equations of the quantum inverse scattering method in the semiclassical limit

D. I. Abramov

Teoretičeskaâ i matematičeskaâ fizika, Tome 63 (1985) no. 1, pp. 32-49

Résumé

The transition to the semiclassicat limit in Marchenko's method for the inverse scattering problem for fixed angular momentum in the $s$-wave case is investigated. It is shown that the kernel $K(r, r')$ of the transformation operator is determined by the classically forbidden region and is exponentially large. Therefore, the linear integral equation for $K(r, r')$ cannot be reduced to a relationship between semiclassical physical quantities. Instead, one uses the equivalent nonlinear equation for the kernel $L(r, r')$ of the inverse operator of the transformation, continued with respect to the first argument to the complete axis. Under semiclassical conditions, the kernel $L(r, r')$ is a rapidly oscillating function having a simple physical meaning, and the nonlinear equation for $L(r, r')$ goes over into the well-known semiclassical relation between the phase shift and the potential. As an example, $s$-wave scattering by an exponential potential is considered.