# Smoothness of Solutions for Schrödinger Equations with Unbounded Potentials

### Shin-ichi Doi

Osaka University, Japan

## 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* ∈ (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 potential is subquadratic. Our method regards the growing potentials as perturbations; so it is applicable to matrix potentials as well.