Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter. (English) Zbl 1533.37132
The author considers two linear Hamiltonian differential systems with Dirichlet boundary conditions and depending on a parameter \(\lambda\). The relative oscillation theorem relates the difference of the numbers of finite eigenvalues of the two problems in the intervals \((-\infty, \beta]\) and \((-\infty, \alpha]\) with the so-called oscillation numbers associated with the Wronskian of the principal solutions of the systems evaluated for \(\lambda=\alpha\) and \(\lambda=\beta\). For the proofs the author drops the assumptions on controllability and normality and also omits the Legendre condition (convexity of the quadratic form in the Legendre transformation).
As a corollary to the general main result the author proves the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in \((\alpha, \beta]\).
The article is written in a straightforward way without too many explanations and requires a quite high level of familiarity with the used concepts (index theory, (dual) comparative indices).
As a corollary to the general main result the author proves the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in \((\alpha, \beta]\).
The article is written in a straightforward way without too many explanations and requires a quite high level of familiarity with the used concepts (index theory, (dual) comparative indices).
Reviewer: Alois Steindl (Wien)
MSC:
| 37J51 | Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods |
| 37J06 | General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
Keywords:
comparative index; finite eigenvalues; linear Hamiltonian systems; relative and renormalized oscillation theory; spectral and oscillation theoryReferences:
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