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

TL;DR

This paper establishes the well-posedness of non-autonomous linear evolution equations with generators having scalar commutators, providing explicit solutions and relaxing continuity assumptions, with applications to quantum operators.

Contribution

It extends well-posedness results to generators with scalar commutators and relaxes norm continuity to strong continuity, including explicit solution formulas.

Findings

Proves well-posedness for generators with scalar pairwise commutators.

Provides explicit representation formulas for the evolution.

Relaxes Kato's norm continuity condition to strong continuity.

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 $1$-fold 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 Schr\"odinger operators for particles in external electric fields.