  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 $(t,x)\in[0,T]\times\R^n$ and complex valued. Under a suitable
decay condition as $|x|\to\infty$ on the imaginary part of the first order term
and an algebraic growth assumption on the real part, we derive global energy
estimates in suitable Sobolev spaces of infinite order and prove a well
posedness result in Gelfand-Shilov type spaces. We also discuss by examples the
sharpness of the result.
