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

The semiclassical asymptotics of the solution to the Cauchy problem for the nonstationary Schrödinger equation $$\hat{L} = -i\hbar\frac{\partial}{\partial t} - \frac{\hbar^2}{2}\Delta + \alpha \delta_{x_0},\quad \alpha \in \mathbb{R},$$ 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.