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@article{1705958,
author = {Šepitka, Peter and Šimon Hilscher, Roman},
article_number = {February 2021},
doi = {http://dx.doi.org/10.1016/j.laa.2020.11.018},
keywords = {Linear Hamiltonian system; Left focal point; Right focal point; Comparative index; Principal solution; Sturmian theory},
language = {eng},
issn = {0024-3795},
journal = {Linear Algebra and Its Applications},
title = {Distribution and number of focal points for linear Hamiltonian systems},
url = {https://doi.org/10.1016/j.laa.2020.11.018},
volume = {611},
year = {2021}
}
TY - JOUR
ID - 1705958
AU - Šepitka, Peter - Šimon Hilscher, Roman
PY - 2021
TI - Distribution and number of focal points for linear Hamiltonian systems
JF - Linear Algebra and Its Applications
VL - 611
IS - February 2021
SP - 26-45
EP - 26-45
PB - Elsevier
SN - 00243795
KW - Linear Hamiltonian system
KW - Left focal point
KW - Right focal point
KW - Comparative index
KW - Principal solution
KW - Sturmian theory
UR - https://doi.org/10.1016/j.laa.2020.11.018
L2 - https://doi.org/10.1016/j.laa.2020.11.018
N2 - In this paper we consider the question of distribution and number of left and right focal points for conjoined bases of linear Hamiltonian differential systems. We do not assume any complete controllability (identical normality) condition. Recently we obtained the Sturmian separation theorem for this case which provides optimal lower and upper bounds for the numbers of left and right focal points of every conjoined basis in terms of the principal solutions at the endpoints of the interval. In this paper we show that for any two given integers within these bounds there exists a conjoined basis with these prescribed numbers of left and right focal points. We determine such conjoined bases by their initial conditions. Our approach is to transfer the problem through the comparative index into matrix analysis. The main results are new even for completely controllable linear Hamiltonian systems. As an application we extend a classical result for controllable systems by Reid (1971) about the existence of conjoined bases with an invertible first component.
ER -
ŠEPITKA, Peter and Roman ŠIMON HILSCHER. Distribution and number of focal points for linear Hamiltonian systems. \textit{Linear Algebra and Its Applications}. Elsevier, 2021, vol.~611, February 2021, p.~26-45. ISSN~0024-3795. Available from: https://dx.doi.org/10.1016/j.laa.2020.11.018.
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