nLab Magnus expansion

Idea

The Magnus expansion is a continuous analogue of the Hausdorff series in Lie theory. It is the basis for some methods applying Lie theory to non-autonomous linear differential equations for linear operators.

Related concepts

• Dyson series

Literature

The Magnus expansion was proposed in

• Wilhelm Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7 4 (1954) 649–673

  (1954) [doi:10.1002/cpa.3160070404]

Surveys:

• S. Blanes, F. Casas, J. A. Oteo, J. Ros, The Magnus expansion and some of its application, Physics reports 470 (2008) 151–238 doi

• Kurusch Ebrahimi-Fard, Igor Mencattini, Alexandre Quesney, What is the Magnus Expansion?, arXiv:2312.16674

Relation to the Dyson series:

• Michel Bauer, Raphael Chetrite, Kurusch Ebrahimi-Fard, Frederic Patras: Time-ordering and a generalized Magnus expansion, Letters in Mathematical Physics 103 (2013) 331 [doi:10.1007/s11005-012-0596-z, arXiv:1206.3990]

Path signature generalized to surface signature (for surface iterated integrals, higher parallel transport) and related to Lie algebra crossed modules:

• Ilya Chevyrev, Joscha Diehl, Kurusch Ebrahimi-Fard, Nikolas Tapia, A multiplicative surface signature through its Magnus expansion, arXiv:2406.16856

A post-Lie algebra version

• M. J. H. Al-Kaabi, K. Ebrahimi-Fard, D. Manchon, Post-Lie Magnus expansion and BCH-recursion, SIGMA 18, (2022), 023, 16 pages.