2014/11/04 by Jochen Schmid, Schmid, Jochen
Computer Science · Mathematics · #35Q41 (secondary) #47D06 (primary) #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics #math.AP #msc:35Q41 #msc:47D06
paper · pdf · doi:10.48550/arxiv.1411.0857
30 pages, to appear in the Journal of Evolution Equations
openalex publication_date 2014/11/04 · arxiv created 2015/06/23 · arxiv updated 2015/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the well-posedness of non-autonomous linear evolution equations for generators A(t): D(A(t)) ⊂ X → 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 Schrödinger operators for particles in external electric fields.