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

We study the Schrodinger equation i partial derivative(t)u + Delta u + V(0)u + V(1)u = 0 on R-3 x (0, T), where V-0(x, t) = |x - a(t)|(-1), with a is an element of W-2,W-1 (0, T; R-3), is a coulombian potential, singular at finite distance, and VI is an electric potential, possibly unbounded. The initial condition u(0) is an element of H-2(R-3) is such that f(R3) (1 + |x|(2))(2)|u(0)(x)|(2)dx < infinity. 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. (c) 2005 Elsevier Inc. All rights reserved.