Sufficient conditions for the convergence of the Magnus expansion

TL;DR

This paper presents two new sufficient conditions for the convergence of the Magnus expansion in solving linear differential equations, one based on operator norms and the other on spectral properties, with illustrative examples.

Contribution

It introduces novel convergence criteria for the Magnus expansion, enhancing understanding of when it converges in terms of operator norms and spectral structure.

Findings

First condition provides a norm-based convergence bound.

Second condition relates convergence to spectral properties.

Examples illustrate the effectiveness of both conditions.

Abstract

Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation $Y' = A(t) Y$. The first one provides a bound on the convergence domain based on the norm of the operator $A(t)$. The second condition links the convergence of the expansion with the structure of the spectrum of $Y(t)$, thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.