Abstract
In this paper, we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. Our results generalize the known theory of linear Hamiltonian systems in two respects. Namely, we allow nonlinear dependence of the coefficients on the spectral parameter and at the same time we do not impose any controllability and strict normality assumptions. We introduce the notion of a finite eigenvalue and prove the oscillation theorem relating the number of finite eigenvalues which are less than or equal to a given value of the spectral parameter with the number of proper focal points of the principal solution of the system in the considered interval. We also define the corresponding geometric multiplicity of finite eigenvalues in terms of finite eigenfunctions and prove that the algebraic and geometric multiplicities coincide. the results are also new for Sturm-Liouville differential equations, being special linear Hamiltonian systems. © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Recommended Citation
M. Bohner et al., "Oscillation and Spectral Theory for Linear Hamiltonian Systems with Nonlinear Dependence on the Spectral Parameter," Mathematische Nachrichten, vol. 285, no. 11 thru 12, pp. 1343 - 1356, Wiley; Wiley-VCH Verlag, Aug 2012.
The definitive version is available at https://doi.org/10.1002/mana.201100172
Department(s)
Mathematics and Statistics
Keywords and Phrases
Conjoined basis; Controllability; Finite eigenvalue; Linear Hamiltonian system; Normality; Oscillation; Proper focal point; Quadratic functional; Self-adjoint eigenvalue problem
International Standard Serial Number (ISSN)
1522-2616; 0025-584X
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2024 Wiley; Wiley-VCH Verlag, All rights reserved.
Publication Date
01 Aug 2012
