Resolvent estimates for scalar fields with electromagnetic perturbation

In this note we prove some estimates for the resolvent of the operator $-Delta$ perturbed by the differential operator $$ V(x,D)=ia(x)cdot abla+V(x)quad hbox{in }mathbb{R}^3,. $$ This differential operator is of short range type and a compact perturbation of the Laplacian on $mathbb{R}^3$. Also we find estimates in the space-time norm for the solution of the wave equation with such perturbation.