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

• Published in: Communications in mathematical physics Vol. 287; no. 2; pp. 613 - 640
• Main Authors: Krüger, Helge, Teschl, Gerald
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-04-2009; Springer
• Subjects: Classical and Quantum Gravitation; Complex Systems; Exact sciences and technology; Mathematical analysis; Mathematical and Computational Physics; Mathematical methods in physics; Mathematical Physics; Mathematics; Operator theory; Other topics in mathematical methods in physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Sciences and techniques of general use; Theoretical; Trace Class; Essential Spectrum; Jacobi Operator; Liouville Operator; Liouville Theory; Oscillations; Eigenvalues; Mathematical physics
• ISSN: 0010-3616; 1432-0916
• DOI: 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 number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein’s spectral shift function is established.