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

TL;DR

This paper proves the existence of $ ext{ extsterling}$-inverses for certain functions, enabling a Lanczos-like algorithm to evaluate the time-ordered exponential of time-dependent matrices, with applications to Green's function inverse problems.

Contribution

It provides a constructive proof of $ ext{ extsterling}$-inverse existence for smooth, separable functions, advancing algorithms for time-ordered exponentials and inverse Green's function problems.

Findings

$ ext{ extsterling}$-inverses exist for all non-zero smooth, separable functions

The proof enables a Lanczos-like algorithm for time-ordered exponentials

Partial solution to the Green's function inverse problem

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 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.