Schrödinger-type equations in Gelfand-Shilov spaces

We study the initial value problem for Schr"odinger-type equations with initial data presenting a certain Gevrey regularity and an exponential behavior at infinity. We assume the lower order terms of the Schr"odinger operator depending on $(t,x) in [0,T] imes R^n$ and complex valued. Under a suitable decay condition as $|x| o infty$ on the imaginary part of the first order term and an algebraic growth assumption on the real part, we prove a well posedness result and global energy estimates in suitable Gelfand-Shilov type spaces. We also discuss by examples the sharpness of the result.