The time-ordered exponential of a time-dependent matrix $\mathsf{A}(t)$ is defined as the function of $\mathsf{A}(t)$ that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in $\mathsf{A}(t)$. The authors recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted $\ast$. Yet, the existence of such inverses, crucial to avoid algorithmic breakdowns, still needed to be proved. Here we constructively prove that $\ast$-inverses exist for all non-identically null, smooth, separable functions of two variables. As a corollary, we partially solve the Green's function inverse problem which, given a distribution $G$, asks for the differential operator whose fundamental solution is $G$. Our results are abundantly illustrated by examples.

中文翻译：

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时间顺序指数的类 Lanczos 算法：$\ast$-inverse 问题

时间相关矩阵 $\mathsf{A}(t)$ 的时间有序指数定义为 $\mathsf{A}(t)$ 的函数，该函数求解耦合线性微分方程的一阶系统在 $\mathsf{A}(t)$ 中编码的非常数系数。作者最近提出了第一个能够评估该函数的类似 Lanczos 的算法。该算法依赖于与非交换卷积类产品相关的时间相关函数的逆，表示为 $\ast$。然而，这种对避免算法崩溃至关重要的逆的存在仍然需要证明。在这里，我们建设性地证明 $\ast$-inverses 存在于两个变量的所有不相同的 null、平滑、可分离的函数。作为推论，我们部分解决了格林函数逆问题，给定分布 $G$，求基本解为 $G$ 的微分算子。我们的结果通过例子得到了充分的说明。