Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence estimates for the Magnus expansion III. Banach--Lie algebras

Gyula Lakos

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 III, we consider the Banach--Lie algebraic setting. We show how to improve the "standard" convergence bound $\boldsymbolδ= 2.1737374\ldots$ using the customary generating function / ODE methods. Then we discuss how to achieve better convergence bounds using the resolvent method. The emphasis here is on the variety of the methods, rather than pushing them to the extreme. Nevertheless, regarding the cumulative convergence radii, we show how to establish $2.427<\mathrm C_\infty^{\mathrm{Lie}}\leq 4$ for the Magnus expansion, and $2.93<\mathrm C^{\mathrm{Lie}}_2 \leq 2\boldsymbol v_{\mathrm{Mi}}=5.4028570\ldots$ for the BCH expansion.

Convergence estimates for the Magnus expansion III. Banach--Lie algebras

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 III, we consider the Banach--Lie algebraic setting. We show how to improve the "standard" convergence bound using the customary generating function / ODE methods. Then we discuss how to achieve better convergence bounds using the resolvent method. The emphasis here is on the variety of the methods, rather than pushing them to the extreme. Nevertheless, regarding the cumulative convergence radii, we show how to establish for the Magnus expansion, and for the BCH expansion.

Paper Structure

This paper contains 13 sections, 32 theorems, 362 equations, 1 figure.

Table of Contents

  1. The Magnus commutators as Lie polynomials
  2. Principles of convergence
  3. Divergence estimate for the BCH expansion
  4. The standard estimate and its improvements
  5. The extension of the Banach-Lie norm in the formal case
  6. The resolvent method and the Magnus expansion
  7. The spectral estimates
  8. The structure of the kernels
  9. Estimates in low orders
  10. The resolvent method and the BCH expansion
  11. Numerics and heuristics
  12. Magnus commutators and $\Theta^{\mathop{\mathrm{Lie}}\nolimits}_n$ of low degree
  13. The BCH expansion and $\Gamma^{\mathop{\mathrm{Lie}}\nolimits}_{n,m}$ in low degrees

Key Result

Theorem 1.1

If $|X|<\pi$, or $\mathop{\mathrm{sp}}\nolimits(X)\subset\{z\in\mathbb C\,:\,|z|<\pi\}$, or $\mathop{\mathrm{sp}}\nolimits(X)\subset\{z\in\mathbb C\,:\,|\mathop{\mathrm{Re}}\nolimits z|<\pi\}$ , then $|\mathop{\mathrm{ad}}\nolimits X|_{\mathfrak A}<2\pi$, or $\mathop{\mathrm{sp}}\nolimits(\mathop{\m and hold; with the usual $\log$ branch cut along the negative real axis.

Figures (1)

  • Figure :

Theorems & Definitions (90)

  • Theorem 1.1: F. Schur Sch1 (1890), Sch2 , Poincaré PH (1899)
  • proof
  • Corollary 1.2
  • Theorem 1.3: Magnus M, 1954
  • proof : Remark
  • proof
  • Remark 1.4
  • Theorem 1.5: Solomon S, 1968
  • proof
  • proof : Remark
  • ...and 80 more