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

International audienceWe study a Schrödinger equation with a coulombian potential, singular at finite distance, and an electric potential, possibly unbounded. The two potentials are real valued and may depend on space and time variables. We prove that if the electric potential 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