Equations of the quantum inverse scattering method in the semiclassical limit

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.