Applications of Mathematics, Vol. 65, No. 6, pp. 807-827, 2020

Lanczos-like algorithm for the time-ordered exponential: The $\ast$-inverse problem

Pierre-Louis Giscard, Stefano Pozza

Received December 5, 2019.   Published online September 24, 2020.

Abstract:  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.
Keywords:  time-ordering; matrix differential equation; time-ordered exponential; Lanczos algorithm; fundamental solution
Classification MSC:  35A24, 47B36, 65F10, 65D15
DOI:  10.21136/AM.2020.0342-19

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Affiliations:   Pierre-Louis Giscard, University Littoral Côte d'Opale, UR 2597, LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, F-62100 Calais, France, e-mail:; Stefano Pozza, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail:

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