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

Julia Elyseeva

Mathematische Nachrichten, 2023, vol. 296, issue 1, 196-216

Abstract: In this paper, we consider two linear Hamiltonian differential systems that depend in general nonlinearly on the spectral parameter λ 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 (−∞,β]$(-\infty , \beta ]$ and (−∞,α]$(-\infty , \alpha ]$, respectively, with the so‐called oscillation numbers associated with the Wronskian of the principal solutions of the systems evaluated for λ=α$\lambda =\alpha$ and λ=β$\lambda =\beta$. As a corollary to the main result, we prove the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in (α,β]$(\alpha ,\beta ]$. The consideration is based on the comparative index theory applied to the continuous case.

Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1002/mana.202000434

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bla:mathna:v:296:y:2023:i:1:p:196-216

Ordering information: This journal article can be ordered from
http://www.blackwell ... bs.asp?ref=0025-584X

Access Statistics for this article

Mathematische Nachrichten is currently edited by Robert Denk

More articles in Mathematische Nachrichten from Wiley Blackwell
Bibliographic data for series maintained by Wiley Content Delivery ().