Effective Pruefer Angles and Relative Oscillation Criteria

TL;DR

This paper introduces a new approach to oscillation criteria for Sturm-Liouville operators using effective Pruefer angles, enabling precise determination of eigenvalue insertion into spectral gaps.

Contribution

It develops a novel scale of oscillation criteria based on Pruefer angles, unifying and extending classical and recent spectral results for Sturm-Liouville operators.

Findings

Recovered the Gesztesy-Uenal criterion and classical criteria by Kneser, Hartman, Hille, Weber.

Extended Rofe-Beketov results and Schmidt's extensions.

Provided a unified framework for eigenvalue insertion into spectral gaps.

Abstract

We present a streamlined approach to relative oscillation criteria based on effective Pruefer angles adapted to the use at the edges of the essential spectrum. Based on this we provided a new scale of oscillation criteria for general Sturm-Liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap. As a special case we recover the Gesztesy-Uenal criterion (which works below the spectrum and contains classical criteria by Kneser, Hartman, Hille, and Weber) and the well-known results by Rofe-Beketov including the extensions by Schmidt.