Robert Stadler, Gerald Teschl
Relative Oscillation Theory for Dirac Operators
Preprint series: ESI preprints

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory

34B24 Sturm-Liouville theory, See also {34Lxx}

34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions

34L05 General spectral theory

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.


Keywords: oscillation theory, Dirac operators, spectral theory