Lidskii angles and Sturmian theory for linear Hamiltonian
systems on compact interval

J 2021

Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval

ŠEPITKA, Peter and Roman ŠIMON HILSCHER

Basic information

Original name

Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval

Authors

ŠEPITKA, Peter (703 Slovakia, belonging to the institution) and Roman ŠIMON HILSCHER (203 Czech Republic, guarantor, belonging to the institution)

Edition

Journal of Differential Equations, Elsevier, 2021, 0022-0396

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

is not subject to a state or trade secret

References:

URL

Impact factor

Impact factor: 2.615

RIV identification code

RIV/00216224:14310/21:00119050

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1016/j.jde.2021.06.037

UT WoS

000681321100001

EID Scopus

2-s2.0-85109165913

Keywords in English

Linear Hamiltonian system; Lidskii angle; Focal point; Principal solution; Sturmian separation theorem; Limit theorem

Abstract

In the original language

In this paper we investigate the Sturmian theory for general (possibly uncontrollable) linear Hamiltonian systems by means of the Lidskii angles, which are associated with a symplectic fundamental matrix of the system. In particular, under the Legendre condition we derive formulas for the multiplicities of the left and right proper focal points of a conjoined basis of the system, as well as the Sturmian separation theorems for two conjoined bases of the system, in terms of the Lidskii angles. The results are new even in the completely controllable case. As the main tool we use the limit theorem for monotone matrix-valued functions by Kratz (1993). The methods allow to present a new proof of the known monotonicity property of the Lidskii angles. The results and methods can also be potentially applied in the singular Sturmian theory on unbounded intervals, in the oscillation theory of linear Hamiltonian systems without the Legendre condition, in the comparative index theory, or in linear algebra in the theory of matrices.

Links

GA19-01246S, research and development project

Name: Nová oscilační teorie pro lineární hamiltonovské a symplektické systémy

Investor: Czech Science Foundation

Cite

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval. Journal of Differential Equations. Elsevier, 2021, vol. 298, October, p. 1-29. ISSN 0022-0396. Available from: https://dx.doi.org/10.1016/j.jde.2021.06.037.

@article{1781698,
   author = {Šepitka, Peter and Šimon Hilscher, Roman},
   article_number = {October},
   doi = {http://dx.doi.org/10.1016/j.jde.2021.06.037},
   keywords = {Linear Hamiltonian system; Lidskii angle; Focal point; Principal solution; Sturmian separation theorem; Limit theorem},
   language = {eng},
   issn = {0022-0396},
   journal = {Journal of Differential Equations},
   title = {Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval},
   url = {https://doi.org/10.1016/j.jde.2021.06.037},
   volume = {298},
   year = {2021}
}

TY  - JOUR
ID  - 1781698
AU  - Šepitka, Peter - Šimon Hilscher, Roman
PY  - 2021
TI  - Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval
JF  - Journal of Differential Equations
VL  - 298
IS  - October
SP  - 1-29
EP  - 1-29
PB  - Elsevier
SN  - 00220396
KW  - Linear Hamiltonian system
KW  - Lidskii angle
KW  - Focal point
KW  - Principal solution
KW  - Sturmian separation theorem
KW  - Limit theorem
UR  - https://doi.org/10.1016/j.jde.2021.06.037
N2  - In this paper we investigate the Sturmian theory for general (possibly uncontrollable) linear Hamiltonian systems by means of the Lidskii angles, which are associated with a symplectic fundamental matrix of the system. In particular, under the Legendre condition we derive formulas for the multiplicities of the left and right proper focal points of a conjoined basis of the system, as well as the Sturmian separation theorems for two conjoined bases of the system, in terms of the Lidskii angles. The results are new even in the completely controllable case. As the main tool we use the limit theorem for monotone matrix-valued functions by Kratz (1993). The methods allow to present a new proof of the known monotonicity property of the Lidskii angles. The results and methods can also be potentially applied in the singular Sturmian theory on unbounded intervals, in the oscillation theory of linear Hamiltonian systems without the Legendre condition, in the comparative index theory, or in linear algebra in the theory of matrices.
ER  -

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. Lidskii angles and Sturmian theory for linear Hamiltonian systems on compact interval. \textit{Journal of Differential Equations}. Elsevier, 2021, vol.~298, October, p.~1-29. ISSN~0022-0396. Available from: https://dx.doi.org/10.1016/j.jde.2021.06.037.

Přehled o publikaci