Relative Oscillation Theory for Dirac Operators

TL;DR

This paper introduces a relative oscillation theory for Dirac operators, enabling comparison of spectral differences between two operators and analyzing eigenvalues in spectral gaps using Wronskians and spectral shift functions.

Contribution

It develops a new relative oscillation framework for Dirac operators, extending classical oscillation theory to compare spectra of different operators and connect with spectral shift functions.

Findings

Sturm-type comparison theorem holds for relative oscillation.

Method to determine eigenvalues in spectral gaps.

Extension of results on eigenvalue finiteness in spectral gaps.

Abstract

We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.