A Unifying Framework for Perturbative Exponential Factorizations

Abstract

We propose a framework where Fer and Wilcox expansions for the solution of differential equations are derived from two particular choices for the initial transformation that seeds the product expansion. In this scheme, intermediate expansions can also be envisaged. Recurrence formulas are developed. A new lower bound for the convergence of the Wilcox expansion is provided, as well as some applications of the results. In particular, two examples are worked out up to a high order of approximation to illustrate the behavior of the Wilcox expansion.

Keywords

• sequences of linear transformations
• Wilcox expansion
• Fer expansion
• Zassenhaus formula
• Bellman problem