Helge Krüger, Gerald Teschl
Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function
The paper is published: Comm. Math. Phys. 287:2 (2009) 613-640

MSC:

34B24 Sturm-Liouville theory, See also {34Lxx}

34C10 Oscillation theory, zeros, disconjugacy and comparison theory

34L15 Estimation of eigenvalues, upper and lower bounds

34L05 General spectral theory

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: Sturm--Liouville operators, oscillation theory, spectral shift function
Notes: second and final version