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

We deal with the following Cauchy problem for a Schrödinger equation: Dtu-Δu+∑j=1naj(t,x)Dxju+b(t,x)u=0,u(0,x)=g(x).We assume a decay condition of type | x| -σ, σ∈ (0 , 1) , on the imaginary part of the coefficients aj of the convection term for large values of |x|. This condition is known to produce a unique solution with Gevrey regularity of index s≥ 1 and loss of an infinite number of derivatives with respect to the data for every s≤11-σ. In this paper, we consider the case s>11-σ, 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 Hm, m≥ 0. As a by-product, we show that a decay condition on data in Hm produces a solution with (at least locally) the same regularity as the data, but with an expected different behavior as | x| → ∞.