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

We study the Schrödinger equation i∂tu+Δu+V0u+V1u=0 on R3×(0,T), where V0(x,t)=|x-a(t)|-1, with a∈W2,1(0,T;R3), is a coulombian potential, singular at finite distance, and V1 is an electric potential, possibly unbounded. The initial condition u0∈H2(R3) is such that ∫R3(1+|x|2)2|u0(x)|2dx<∞. The potential V1 is also real valued and may depend on space and time variables. We prove that if V1 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.