On the absolutely continuous spectrum of generalized indefinite strings

Abstract

We investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the conservative Camassa-Holm flow in the dispersive regime is essentially supported on the interval [1/4, ∞).

Quantum graphs on radially symmetric antitrees

“…Proof. As in the proof of Theorem 9.1, we know that the operator H is unitarily equivalent to the operator H. By Theorem 3.1 from [14],…”

“…However, spectral theory of strings enables us to treat more subtle problems (like the study of the structure of the essential spectrum of H). In Section 9, we employ the recent results from [4] and [14] on the absolutely continuous spectrum of strings to construct several classes of antitrees with absolutely continuous spectrum supported on [0, ∞). For instance, if…”

“…The results of the previous section show that antitrees with nonempty absolutely continuous spectrum is a rare event. Our main aim in this section is to apply two recent result from [4] and [14] on the absolutely continuous spectrum of Krein and generalized indefinite strings, respectively, in order to construct several classes of antitrees with rich absolutely continuous spectra, however, which are not eventually periodic in the sense of Theorem 8.3. We begin with the following result.…”

On the absolutely continuous spectrum of generalized indefinite strings II

“…In this paper, we continue our study of the absolutely continuous part of the spectrum of generalized indefinite strings initiated in [17]. We recall briefly (see Section 2 for further details) that a generalized indefinite string is a triple (L, ω, υ) such that L ∈ (0, ∞], ω is a real distribution in H −1 loc [0, L) and υ is a non-negative Borel measure on the interval [0, L).…”

“…The main reason for this lies in the fact that the spectral parameter appears in the wrong place, which does not allow to view (1.4) as an additive perturbation immediately. Our approach to the absolutely continuous spectrum of generalized indefinite strings follows [17] and is based on the elegant ideas of Deift and Killip [9]. In order to implement this approach, we need two main ingredients: The first ingredient is a continuity property for the correspondence between generalized indefinite strings and their associated Weyl-Titchmarsh functions (see Section 2 for more details) obtained in [14,Proposition 6.2].…”