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@article{1353395,
author = {Dřímalová, Iva and Kratz, Werner and Šimon Hilscher, Roman},
article_location = {HEIDELBERG},
article_number = {3},
doi = {http://dx.doi.org/10.1007/s10231-016-0611-6},
keywords = {Sturm-Liouville differential equation; Linear Hamiltonian system; Generalized quasiderivative; Oscillation theory; Spectral theory; Quadratic functional; Rayleigh principle},
language = {eng},
issn = {0373-3114},
journal = {Annali di Matematica Pura ed Applicata. Series IV},
title = {Sturm-Liouville matrix differential systems with singular leading coefficient},
volume = {196},
year = {2017}
}
TY - JOUR
ID - 1353395
AU - Dřímalová, Iva - Kratz, Werner - Šimon Hilscher, Roman
PY - 2017
TI - Sturm-Liouville matrix differential systems with singular leading coefficient
JF - Annali di Matematica Pura ed Applicata. Series IV
VL - 196
IS - 3
SP - 1165-1183
EP - 1165-1183
PB - Springer HEIDELBERG
SN - 03733114
KW - Sturm-Liouville differential equation
KW - Linear Hamiltonian system
KW - Generalized quasiderivative
KW - Oscillation theory
KW - Spectral theory
KW - Quadratic functional
KW - Rayleigh principle
N2 - In this paper we study a general even order symmetric Sturm-Liouville matrix differential equation, whose leading coefficient may be singular on the whole interval under consideration. Such an equation is new in the current literature, as it is equivalent with a system of Sturm-Liouville equations with different orders. We identify the so-called normal form of this equation, which allows to transform this equation into a standard (controllable) linear Hamiltonian system. Based on this new transformation we prove that the associated eigenvalue problem with Dirichlet boundary conditions possesses all the traditional spectral properties, such as the equality of the geometric and algebraic multiplicities of the eigenvalues, orthogonality of the eigenfunctions, the oscillation theorem and Rayleigh's principle, and the Fourier expansion theorem. We also discuss sufficient conditions, which allow to reduce a general even order symmetric Sturm-Liouville matrix differential equation into the normal form. Throughout the paper we provide several examples, which illustrate our new theory.
ER -
DŘÍMALOVÁ, Iva, Werner KRATZ and Roman ŠIMON HILSCHER. Sturm-Liouville matrix differential systems with singular leading coefficient. \textit{Annali di Matematica Pura ed Applicata. Series IV}. HEIDELBERG: Springer HEIDELBERG, 2017, vol.~196, No~3, p.~1165-1183. ISSN~0373-3114. Available from: https://dx.doi.org/10.1007/s10231-016-0611-6.
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