Smoothness of Solutions for Schrödinger Equations with Unbounded Potentials

We consider a Schrodinger equation with linearly bounded magnetic potentials and a quadratically bounded electric potential when the coefficients of the principal part do not necessarily converge to constants near infinity. Assuming that there exists a suitable function f (x) near infinity which is convex with respect to the Hamilton vector field generated by the (scalar) principal symbol, we show a microlocal smooth- ing effect, which says that the regularity of the solution increases for all time t ∈ (0 ,T ) at every point that is not trapped backward by the geodesic flow if the initial data decays in an incoming region in the phase space. Here T depends on the potentials; we can choose T = ∞ if the magnetic potentials are sublinear and the electric poten- tial is subquadratic. Our method regards the growing potentials as perturbations; so it is applicable to matrix potentials as well.