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

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

2007-03-20

arxiv.org/abs/math/0703574v2

Commun. Math. Phys. 287:2, 613-640 (2009)

Mathematics

Spectral Theory

26 pages

Scientific paper

10.1007/s00220-008-0600-8

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.

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