# Lanczos-like algorithm for the time-ordered exponential: The ∗-inverse problem

The time-ordered exponential of a time-dependent matrix A(t) is defined as the function of A(t) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in 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 ∗. Yet, the existence of such inverses, crucial to avoid algorithmic breakdowns, still needed to be proved. Here we constructively prove that ∗-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.

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