Article

**Commun. Math. Phys. 287:2, 613-640 (2009)**[DOI: 10.1007/s00220-008-0600-8]

## Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

### Helge Krüger and Gerald Teschl

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 number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.

** MSC2000:** Primary 34B24, 34C10; Secondary 34L15, 34L05

**Keywords:** *Sturm-Liouville operators, oscillation theory, spectral shift function*

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