Some remarks on the Cauchy Problem for Schr\"odinger type equations in Gelfand-Shilov spaces

TL;DR

This paper investigates the well-posedness of the Cauchy problem for Schrödinger type equations within Gelfand-Shilov spaces, establishing conditions under which solutions exist, are unique, and depend continuously on initial data.

Contribution

It provides new well-posedness results for Schrödinger equations in Gelfand-Shilov spaces under decay assumptions, and discusses the optimality of these conditions with examples.

Findings

Well-posedness established under decay conditions on coefficients

Optimality of conditions demonstrated through examples

Extension of classical results to Gelfand-Shilov spaces

Abstract

We consider the Cauchy problem for Schr\"odinger type operators. Under a suitable decay assumption on the imaginary part of the first order coefficients we prove well-posedness of the Cauchy problem in Gelfand-Shilov classes. We also discuss the optimality of our result through some examples.