Relative oscillation theory for Sturm–Liouville operators extended

Abstract

We extend relative oscillation theory to the case of Sturm-Liouville operators H u = r −1 (−(pu ) + qu) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.

Effective Prüfer angles and relative oscillation criteria

“…One can interpret # (c,d) (u 0 , u 1 ) as the weighted sign flips of the Wronskian W x (u 0 , u 1 ), where a sign flip is counted as +1 if q 0 − q 1 and p 0 − p 1 are positive in a neighborhood of the sign flip, it is counted as −1 if q 0 − q 1 and p 0 − p 1 are negative in a neighborhood of the sign flip. In the case where the differences vanish or are of opposite sign are more subtle [12,13].…”

On perturbations of quasiperiodic Schrödinger operators

“…It was already shown in Corollary 6.6 in [8] that μ crit (E) has to diverge as E → ∞. We also remark that [8] develops a different approach to relative oscillation criteria than was used in [12].…”

“…From Theorem 2.10 of [12], we have that Theorem 3.3. Assume the assumptions of the last theorem, and that for some n ∈ N…”

“…It is shown in Lemmas 4.2 and 4.3 of [12], that there exists a Prüfer angle ψ for W (u 0 , u 1 ) such that it obeys…”

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

“…In particular, it was shown by Krüger and Teschl [96] that one can take solutions of different operators if the right-hand side is interpreted as spectral shift between these two operators. 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].…”