Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function
Spectral Theory
2009-03-03 v2 Classical Analysis and ODEs
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.
Keywords
Cite
@article{arxiv.math/0703574,
title = {Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function},
author = {Helge Krueger and Gerald Teschl},
journal= {arXiv preprint arXiv:math/0703574},
year = {2009}
}
Comments
26 pages