Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence estimates for the Magnus expansion I. Banach algebras

Gyula Lakos

TL;DR

The paper advances convergence theory for the Magnus and BCH expansions in Banach algebras by combining combinatorial and resolvent methods. It establishes precise cumulative-norm convergence radii, showing the Magnus expansion converges for $\int |\,\phi\,|<2$ (with boundary cases covered under Lebesgue–Bochner assumptions) and the BCH series converges for $|X|+|Y|<C_2$ with $C_2\approx2.89847930$. It develops an infinitesimal resolvent framework, introduces $\mathcal{R}^{(\lambda)}(A)$ and $\mu_k^{(\lambda)}$, and proves a logarithmic Magnus formula under $M$-controlled conditions, linking time-ordered exponentials to resolvent-based logarithms. The work also presents a holomorphic-calculus approach via resolvents, defines fractional powers $A^{\alpha}$, and formulates canonical projections through chronological decompositions, while a discrete resolvent approach yields exact decompositions of the resolvent for products. Collectively, the results sharpen convergence thresholds, illuminate the structure of Magnus/BCH terms, and connect spectral and combinatorial techniques to time-ordered exponentials in general Banach-algebra contexts.

Abstract

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part I, we consider the general Banach algebraic setting. We show that the (cumulative) convergence radius of the Magnus expansion is $2$; and of the Baker--Campbell--Hausdorff series is $\mathrm C_2=2.89847930\ldots$. More generally, the resolvent method is developed in the analytic setting.

Convergence estimates for the Magnus expansion I. Banach algebras

TL;DR

The paper advances convergence theory for the Magnus and BCH expansions in Banach algebras by combining combinatorial and resolvent methods. It establishes precise cumulative-norm convergence radii, showing the Magnus expansion converges for (with boundary cases covered under Lebesgue–Bochner assumptions) and the BCH series converges for with . It develops an infinitesimal resolvent framework, introduces and , and proves a logarithmic Magnus formula under -controlled conditions, linking time-ordered exponentials to resolvent-based logarithms. The work also presents a holomorphic-calculus approach via resolvents, defines fractional powers , and formulates canonical projections through chronological decompositions, while a discrete resolvent approach yields exact decompositions of the resolvent for products. Collectively, the results sharpen convergence thresholds, illuminate the structure of Magnus/BCH terms, and connect spectral and combinatorial techniques to time-ordered exponentials in general Banach-algebra contexts.

Abstract

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part I, we consider the general Banach algebraic setting. We show that the (cumulative) convergence radius of the Magnus expansion is ; and of the Baker--Campbell--Hausdorff series is . More generally, the resolvent method is developed in the analytic setting.

Paper Structure

This paper contains 24 sections, 110 theorems, 534 equations, 2 figures.

Table of Contents

  1. The combinatorial approach
  2. The Magnus expansion
  3. The Baker--Campbell--Hausdorff expansion
  4. Some algebraic properties of the Magnus commutators
  5. On the power-series based holomorphic calculus
  6. The (infinitesimal) resolvent approach
  7. Introduction to the resolvent method
  8. The logarithmic Magnus formula
  9. Chronological decompositions the Magnus integrand
  10. Holomorphic calculus through the resolvent
  11. The discrete resolvent approach
  12. Exact and formal decompositions of the resolvent
  13. Some algebraic consequences for the BCH expansion
  14. Analytic consequences for the BCH expansion
  15. Other algebraic consequences of the discrete resolvent decomposition
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\phi$ be a continuous $\mathfrak A$-valued measure of finite variation on the interval $I$. If $T$ is a formal commutative variable, then

Figures (2)

  • Figure 1: The sets ${\mathrm C}_\infty^{(\lambda)}=r$ with $\lambda=x+\mathrm iy$ and $r=\frac{1}{2}$, $\boldsymbol{F_1}$, $\sqrt2$, $\boldsymbol2$, $F_2$.
  • Figure 3: The sets ${\mathrm C}_\infty^{\{u\}}=r$ with $u=x+\mathrm iy$ and $r=\frac{1}{2}$, $\boldsymbol{F_1}$, $\sqrt2$, $\boldsymbol2$, $F_2$.

Theorems & Definitions (273)

  • Theorem 1.1: Magnus M (1954), Chen Ch (1957)
  • proof
  • Theorem 1.2: Magnus M (1954), Chen Ch (1957)
  • proof
  • proof : Note
  • Theorem 1.3
  • proof
  • Theorem 1.4
  • proof
  • Remark 1.5
  • ...and 263 more