Sturm Oscillation and Comparison Theorems

Abstract

Abstract. This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.

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

“…Since then many extensions have been made, which are far too numerous to be listed here. Rather we refer to the recent survey by Simon [30] or the recent monograph by Zettl [45].…”

Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday

“…For further recent reviews of these methods we refer to the ones by Fritz [38] and Simon [125]. Again this result has triggered several extensions.…”

On the Removal of Finite Discrete Spectrum by Coefficient Stripping

“…While it appears that this argument is well known to some experts, I am aware of only two places that it appears explicitly in print: in Nikishin's lovely paper [2] and in my review article on Sturm oscillation theory for the Sturm 200th Birthday Conference [5].…”

“…(Indeed, it was this case treated in [2] and [5]; Nikishin doesn't use the terminology "oscillation theorem" but he counts sign changes of certain determinants.) Theorem 1.2.…”