Article
J. Math. Anal. Appl. 371, 638-648 (2010)
[DOI: 10.1016/j.jmaa.2010.05.069]
Relative Oscillation Theory for Dirac Operators
Robert Stadler and Gerald Teschl
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.
MSC2000: Primary 34C10, 34B24; Secondary 34L20, 34L05
Keywords: Oscillation theory, Dirac operators, spectral theory
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