Classical Solutions to a Linear Schrödinger Evolution Equation Involving a Coulomb Potential with a Moving Center of Mass

Abstract

This paper is concerned with Cauchy problems for the linear Schrödinger evolution equation (i(∂/∂t) + Δ + |x – a(t)|^–1 + V₁(x,t))u(x,t) = f(x,t) in R^N × [0,T], subject to initial condition: u(·,0) ∈ H²(R^N) ∩ H₂(R^N), where i := $\sqrt{-1}$, N ≥ 3, T > 0 and a : [0,T] → R^N expresses the center of the Coulomb potential, V₁ and f are another real-valued potential and an inhomogeneous term, respectively, while H₂(R^N) := {v ∈ L²(R^N); |x|²v ∈ L²(R^N)}. We show that under some conditions on V₁ and f the equation has a classical solution u(·) ∈ C¹([0,T]; L²(R^N)) ∩ C([0,T]; H²(R^N) ∩ H₂(R^N)).