Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

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Authors

ELYSEEVA Julia

Year of publication 2023
Type Article in Periodical
Magazine / Source Mathematische Nachrichten
MU Faculty or unit

Faculty of Science

Citation
Web https://doi.org/10.1002/mana.202000434
Doi http://dx.doi.org/10.1002/mana.202000434
Keywords comparative index; finite eigenvalues; linear hamiltonian systems; relative and renormalized oscillation theory; spectral and oscillation theory
Description In this paper, we consider two linear Hamiltonian differential systems that depend in general nonlinearly on the spectral parameter lambda and with Dirichlet boundary conditions. For the Hamiltonian problems, we do not assume any controllability and strict normality assumptions and also omit the classical Legendre condition for their Hamiltonians. The main result of the paper, the relative oscillation theorem, relates the difference of the numbers of finite eigenvalues of the two problems in the intervals (-infinity,beta]$(-\infty , \beta ]$ and (-infinity,alpha]$(-\infty , \alpha ]$, respectively, with the so-called oscillation numbers associated with the Wronskian of the principal solutions of the systems evaluated for lambda=alpha$\lambda =\alpha$ and lambda=beta$\lambda =\beta$. As a corollary to the main result, we prove the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in (alpha,beta]$(\alpha ,\beta ]$. The consideration is based on the comparative index theory applied to the continuous case.
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