Remarks on linear Schrödinger evolution equations with Coulomb potential with moving center

Abstract

This paper is concerned with Cauchy problems for the linear Schrödinger evolution equation:

$$i{\partial }_{t}u(x,t)+\text{Δ}u(x,t)+|x-a(t){|}^{-1}u(x,t)+V_1(x,t)u(x,t)=f(x,t)$$

in ${ℝ}^N\times [0,T]$, subject to initial condition: $u(x,0)=u_0(x)\in H^2({ℝ}^N)\cap H_2({ℝ}^N)$,

where $i:=\sqrt{-1}$, $N\ge 3,a:[0,T]\to {ℝ}^N$ expresses the center of the Coulomb potential, $V_1$ and $f:{ℝ}^N\times [0,T]\to R$ are another potential and an inhomogeneous term while

$$H_2({ℝ}^N):=\{v\in L^2({ℝ}^N);|x{|}^{2}v\in L^2({ℝ}^N)\}.$$

The strong formulation of this problem (with $f\equiv 0$ and $N=3$) has been solved by Baudouin-Kavian-Puel (2005) partly with formal computation. In this paper we reconstruct their argument with rigorous proofs. Moreover, we show that the solution $u$ satises the energy estimate

$$\Vert {\partial }_{t}u(t)\Vert +{\Vert u(t)\Vert }_{H^2\cap H_2}\le C_0({\Vert u_0\Vert }_{H^2\cap H_2}+{\Vert f\Vert }_F),$$

where is a constant depending on $a$, $V_1$ and $T$, while $\parallel f\parallel F$ is some norm of $f$.