Effective Prüfer angles and relative oscillation criteria

Abstract

We present a streamlined approach to relative oscillation criteria based on effective Prüfer 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 and generalize the Gesztesy-Ünal 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.

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

“…We refer to [96,97] and the references therein. The question whether eigenvalues accumulate at the boundary of an essential spectral gap based on these methods is considered in [98,117]. In this respect we should also mention the beautiful result by Fritz andÜnal [78] which gives the most general version of Kneser's theorem.…”

Half-linear Euler differential equations in the critical case

“…It suggests to investigate a similar problem for the more general equation (6). (iii) In [1], motivated by the linear case treated in [7], [8], [9], we have investigated oscillatory properties of the equation…”

On perturbations of quasiperiodic Schrödinger operators

“…It was shown in [3] that the matrix A and then β would satisfy the bound |β| C |E + − E − | for some constant C . Then it was shown in [13], that |E + − E − | ce −γ |m| for some constants c and γ . Hence one should expect K (E)…”