Oscillation and spectral theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

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Authors

BOHNER Martin KRATZ Werner ŠIMON HILSCHER Roman

Year of publication 2012
Type Article in Periodical
Magazine / Source Mathematische Nachrichten
MU Faculty or unit

Faculty of Science

Citation
Doi http://dx.doi.org/10.1002/mana.201100172
Field General mathematics
Keywords Linear Hamiltonian system; Self-adjoint eigenvalue problem; Proper focal point; Conjoined basis; Finite eigenvalue; Oscillation; Controllability; Normality; Quadratic functional
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Description 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.
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