The interplay between decay of the data and regularity of the solution in Schrödinger equations

We deal with the Cauchy problem for a Schrödinger type equation with (t,x) depending coefficients and first order terms of type a_j(t,x)D_{x_j}. We assume a decay condition of type |x|^{-sigma}, sigma in (0,1), on the imaginary part of the coefficients a_j of the convection term for large values of |x|. This condition is known to produce a unique solution with Gevrey regularity of index sgeq 1 and loss of an infinite number of derivatives with respect to the data for every sleq rac{1}{1-sigma}. In this paper we consider the case s> rac{1}{1-sigma}, where, in general, Gevrey ill-posedness holds. We explain how the space where a unique solution exists depends on the decay and regularity of an initial data in H^m, mgeq 0. As a byproduct, we show that a decay condition on data in H^m produces a solution with (at least locally) the same regularity as the data, but with an expected different behavior as |x| goes to infinity.