Schr\"odinger-type equations in Gelfand-Shilov spaces

TL;DR

This paper investigates the well-posedness of Schr"odinger-type equations with Gevrey regularity and exponential decay in Gelfand-Shilov spaces, deriving global energy estimates under specific decay and growth conditions.

Contribution

It establishes well-posedness results for Schr"odinger equations with complex, variable coefficients in Gelfand-Shilov spaces, extending previous analyses to broader function spaces.

Findings

Global energy estimates in Sobolev spaces of infinite order

Well-posedness in Gelfand-Shilov spaces under decay and growth conditions

Examples demonstrating the sharpness of the results

Abstract

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.