Geodesic
Lanczos-like algorithm for the time-ordered exponential: The $\ast $-inverse problem
Applications of Mathematics, Tome 65 (2020) no. 6, pp. 807-827

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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 have 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 by $\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.
DOI : 10.21136/AM.2020.0342-19
Classification : 35A24, 47B36, 65D15, 65F10
Keywords: time-ordering; matrix differential equation; time-ordered exponential; Lanczos algorithm; fundamental solution 
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     title = {Lanczos-like algorithm for the time-ordered exponential: {The} $\ast $-inverse problem},
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Giscard, Pierre-Louis; Pozza, Stefano. Lanczos-like algorithm for the time-ordered exponential: The $\ast $-inverse problem. Applications of Mathematics, Tome 65 (2020) no. 6, pp. 807-827. doi: 10.21136/AM.2020.0342-19

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