Renormalized oscillation theory for Hamiltonian systems

TL;DR

This paper extends renormalized oscillation theory to general Hamiltonian systems with block matrix coefficients, enabling analysis in spectral gaps where traditional methods fail.

Contribution

It generalizes existing oscillation theory to Hamiltonian systems with block matrices, covering Sturm-Liouville and Dirac operators as special cases.

Findings

Applicable to spectral gaps where traditional oscillation theory fails

Includes Sturm-Liouville and Dirac operators with block matrix coefficients

Uses Wronskians of solutions for analysis

Abstract

We extend a result on renormalized oscillation theory, originally derived for Sturm-Liouville and Dirac-type operators on arbitrary intervals in the context of scalar coefficients, to the case of general Hamiltonian systems with block matrix coefficients. In particular, this contains the cases of general Sturm-Liouville and Dirac-type operators with block matrix-valued coefficients as special cases. The principal feature of these renormalized oscillation theory results consists in the fact that by replacing solutions by appropriate Wronskians of solutions, oscillation theory now applies to intervals in essential spectral gaps where traditional oscillation theory typically fails.