Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control

Abstract

We study the Schrödinger equation i ∂ t u + Δ u + V 0 u + V 1 u = 0 on R 3 × ( 0 , T ) , where V 0 ( x , t ) = | x - a ( t ) | - 1 , with a ∈ W 2 , 1 ( 0 , T ; R 3 ) , is a coulombian potential, singular at finite distance, and V 1 is an electric potential, possibly unbounded. The initial condition u 0 ∈ H 2 ( R 3 ) is such that ∫ R 3 ( 1 + | x | 2 ) 2 | u 0 ( x ) | 2 dx < ∞ . The potential V 1 is also real valued and may depend on space and time variables. We prove that if V 1 is regular enough and at most quadratic at infinity, this problem is well-posed and the regularity of the initial data is conserved for the solution. We also give an application to the bilinear optimal control of the solution through the electric potential.