Helge Krueger and Gerald Teschl
Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift function
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ABSTRACT. We develop an analog of classical oscillation theory for Sturm-Liouville 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 finiteness
of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's
spectral shift function is established.