Smoothness of Solutions for Schrödinger Equations with Unbounded Potentials

Abstract

We consider a Schrödinger equation with linearly bounded magnetic potentials and a quadratically bounded electric potential when the coeffcients 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 smoothing effect, which says that the regularity of the solution increases for all time $t\in (0,T]$ 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 $T=\infty$ if the magnetic potentials are sublinear and the electric potential is subquadratic. Our method regards the growing potentials as perturbations; so it is applicable to matrix potentials as well.