A unifying framework for perturbative exponential factorizations

TL;DR

The paper addresses solving linear time-dependent systems $\dot{x}=A(t)x$ by unifying two classic exponential-factorization approaches, Fer and Wilcox, within a single framework of sequential exponential transformations. By introducing a seed and a parameter $\lambda$, it derives recursive constructions that generate Wilcox and Fer expansions, provides explicit expressions for the higher-order exponents $W_n(t)$ (via Dyson-type integrals and permutation-based Hopf algebra), and establishes a practical convergence bound $\xi_W \approx 0.65846$. It also connects Wilcox to the Zassenhaus formula as a continuous analogue and extends the method to expand $e^{A+\varepsilon B}$ (Wilcox–Bellman expansion), with detailed SU(2) and SO(3) examples to illustrate behavior at high order. Collectively, the work offers a flexible, high-order perturbative toolkit for time-ordered matrix exponentials in quantum and control applications, clarifying relationships among major exponential factorization schemes and enabling problem-tailored intermediate expansions.

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 high order of approximation to illustrate the behavior of the Wilcox expansion.

Figures (5)

• Figure 1: $D_n$ as a function of $1/n$, and linear extrapolation (red line).
• Figure 2: Accuracy of Wilcox--Bellman product expansion up to order eleven as a function of the ratio $\varepsilon/a$, with $a=1$. The quantity plotted is the squared modulus of the non--diagonal element of the matrix. The vertical grey line stands for the convergence lower bound $\varepsilon=0.658$.
• Figure 3: Absolute error of the approximations given by curves in Figure \ref{['fig:wb1-11']} with $a=1$. The vertical grey line is located at the value of the convergence lower bound $\varepsilon=0.658$.
• Figure 4: Error in the approximations to the matrix trace as a function of the rotation angle $\theta$, for two values $\alpha=\pi/2$ and $\pi/4$.
• Figure 5: Error in the approximations to the matrix trace as a function of the rotation angle $\theta$, for two values $\alpha=3\pi/4$ and $\pi$.