Schr\"odinger-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 $(t,x)\in[0,T]\times\R^n$ 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.