Lev Kapitanski and Igor Rodnianski
Regulated smoothing for Schr\"odinger evolution
(35K, AMSTeX)

ABSTRACT.  We study the smoothing of solutions of time-dependent 
multidimensional Schr\"odinger equations with growing 
at infinity potentials.  We show that if the potential 
grows slower than quadratically, then 
the faster the initial condition $\psi(0,x)$ decays at 
infinity the more regular the wavefunction $\psi(t,x)$ 
is for $t>0$. In particular, the fundamental solution 
is infinitely differentiable for $t>0$. We also prove 
analogous  results for the Schr\"odinger equation with 
magnetic field.