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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 a 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, roč.~196, č.~3, s.~1165-1183. ISSN~0373-3114. Dostupné z: https://dx.doi.org/10.1007/s10231-016-0611-6.
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