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 L̂ = -i∂/∂ t - ²/2Δ + α δx₀, α ∈ R, with a point δ -potential in R³ 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 < |x₀| . Explicit asymptotic formulas describing the evolution of the wave packet and its scattering by the δ -potential are obtained. It is proved that, for the considered class of initial conditions, the scattered wave is purely outgoing.