We consider the Cauchy problem for the time dependent Schrödinger equation with magnetic potential A(t,x) and electrostatic potential V(t,x), with x in Rd. We show that the problem is well-posed in L 2(Rd ) if, as |x|→∞, |A(t,x)| is at most linearly increasing, |V(t,x)| at most quadratically increasing, and if the local singularities of V are at most of type c|x|-2 with small constants |c|. We also claim that these conditions are almost optimal as far as the behaviors at infinity and the local singularities of V are concerned. We show that the Cauchy problem is also well posed on Σ(2), the domain of the harmonic oscillator, if the local singularities of V are slightly less singular.