Semiclassical Asymptotics of the Solution to the Cauchy Problem for the Three-Dimensional Schrödinger Equation with a Point Potential

Abstract

The semiclassical asymptotics of the solution to the Cauchy problem for the nonstationary Schrödinger equation

with a point \(\delta\)-potential in \(\mathbb{R}^3\) is described. The initial condition is taken in the form of a rapidly oscillating wave packet with compact support contained in a ball of sufficiently small radius \(R < |x_0|\). Explicit asymptotic formulas describing the evolution of the wave packet and its scattering by the \(\delta\)-potential are obtained. It is proved that, for the considered class of initial conditions, the scattered wave is purely outgoing.