Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar

Abstract

We prove the well-posedness of non-autonomous linear evolution equations for generators $${A(t): D(A(t)) \subset X \to X}$$ whose pairwise commutators are complex scalars, and in addition, we establish an explicit representation formula for the evolution. We also prove well-posedness in the more general case where instead of the onefold commutators only the p-fold commutators of the operators A(t) are complex scalars. All these results are furnished with rather mild stability and regularity assumptions: Indeed, stability in X and strong continuity conditions are sufficient. Additionally, we improve a well-posedness result of Kato for group generators A(t) by showing that the original norm continuity condition can be relaxed to strong continuity. Applications include Segal field operators and Schrodinger operators for particles in external electric fields.