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 $$[1/4,\infty )$$ [ 1 / 4 , ∞ ) .

Generalized indefinite strings with purely discrete spectrum

“…where z is a complex spectral parameter. Just like for generalized indefinite strings, this differential equation has to be understood in a weak sense in general (we refer to [9], [11] and [13,Section 7] for further details). The differential equation (6.2) gives rise to a self-adjoint linear relation T in the Hilbert space…”

“…More details can be found in [3], [8] and [11,Subsection 4.1] in particular. We have shown in [13,Section 7] that it is possible to transform the differential equation (6.2) into the differential equation for a generalized indefinite string (∞, ω, υ) defined as follows: With the diffeomorphism s : [0, ∞) → [0, ∞) given by…”

“…e −x dυ(x) (6.7) for all Borel sets B ⊆ [0, ∞). It then follows from [13,Section 7] that the spectrum of the linear relation T coincides with the spectrum of the generalized indefinite string (∞, ω, υ). From the criteria in Section 5, we thus readily obtain similar criteria for the spectrum of T to be discrete and to satisfy λ∈σ(T)…”

“…The spectral properties of H χ turn out to be closely connected to the generalized indefinite string S χ = (∞, ω χ , 0); see [13,Lemma 8.1].…”