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- Title
Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function.
- Authors
Krüger, Helge; Teschl, Gerald
- 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 number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein’s spectral shift function is established.
- Subjects
OSCILLATIONS; OSCILLATION theory of functional differential equations; WRONSKIAN determinant; DIFFERENTIAL equations; OSCILLATION theory of differential equations
- Publication
Communications in Mathematical Physics, 2009, Vol 287, Issue 2, p613
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-008-0600-8