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

Pierre-Louis Giscard; Stefano Pozza

Applications of Mathematics (2020)

  • Volume: 65, Issue: 6, page 807-827
  • ISSN: 0862-7940

Abstract

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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 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 * . 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.

How to cite

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Giscard, Pierre-Louis, and Pozza, Stefano. "Lanczos-like algorithm for the time-ordered exponential: The $\ast $-inverse problem." Applications of Mathematics 65.6 (2020): 807-827. <http://eudml.org/doc/297223>.

@article{Giscard2020,
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.},
author = {Giscard, Pierre-Louis, Pozza, Stefano},
journal = {Applications of Mathematics},
keywords = {time-ordering; matrix differential equation; time-ordered exponential; Lanczos algorithm; fundamental solution},
language = {eng},
number = {6},
pages = {807-827},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Lanczos-like algorithm for the time-ordered exponential: The $\ast $-inverse problem},
url = {http://eudml.org/doc/297223},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Giscard, Pierre-Louis
AU - Pozza, Stefano
TI - Lanczos-like algorithm for the time-ordered exponential: The $\ast $-inverse problem
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 6
SP - 807
EP - 827
AB - 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.
LA - eng
KW - time-ordering; matrix differential equation; time-ordered exponential; Lanczos algorithm; fundamental solution
UR - http://eudml.org/doc/297223
ER -

References

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