| Title:
|
Lanczos-like algorithm for the time-ordered exponential: The $\ast $-inverse problem (English) |
| Author:
|
Giscard, Pierre-Louis |
| Author:
|
Pozza, Stefano |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
65 |
| Issue:
|
6 |
| Year:
|
2020 |
| Pages:
|
807-827 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
time-ordering |
| Keyword:
|
matrix differential equation |
| Keyword:
|
time-ordered exponential |
| Keyword:
|
Lanczos algorithm |
| Keyword:
|
fundamental solution |
| MSC:
|
35A24 |
| MSC:
|
47B36 |
| MSC:
|
65D15 |
| MSC:
|
65F10 |
| idZBL:
|
Zbl 07285958 |
| idMR:
|
MR4191370 |
| DOI:
|
10.21136/AM.2020.0342-19 |
| . |
| Date available:
|
2020-11-18T09:39:39Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148398 |
| . |
| Reference:
|
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