On the Absolutely Continuous Spectrum of Generalized Indefinite Strings

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 , ∞ ) .

Absolutely Continuous Spectrum for Quantum Trees

We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum consists of bands of purely absolutely continuous spectrum, along with a discrete set of eigenvalues. Afterwards, we study random perturbations of such trees, at the level of edge length and coupling, and prove the stability of pure AC spectrum, along with resolvent estimates.

Scattering of particles bounded to an infinite planar curve

Non-relativistic quantum particles bounded to a curve in [Formula: see text] by attractive contact [Formula: see text]-interaction are considered. The interval between the energy of the transversal bound state and zero is shown to belong to the absolutely continuous spectrum, with possible embedded eigenvalues. The existence of the wave operators is proved for the mentioned energy interval using the Hamiltonians with the interaction supported by the straight lines as the free ones. Their completeness is not proved. The curve is assumed [Formula: see text]-smooth, non-intersecting, unbounded, asymptotically approaching two different half-lines (non-parallel or parallel but excluding the “U-case”). Physically, the system can be considered as a model of long nanostructural channel.

On singular spectrum of finite dimensional perturbations (to the Aronszajn-Donoghue-Kac theory)

The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations à of an operator A0 = A0* defined on a Hilbert space H. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions AB of a symmetric operator A acting in H in terms of the Weyl function M(·) of the pair {A, A0} and boundary n-dimensional operator B = B*. Assuming that the multiplicity of singular spectrum of A0 is maximal it is established orthogonality of singular parts EsAв and EsAo of the spectral measures EAв and EAo of the operators AB and A0, respectively. It is shown that the multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. In particular, it is obtained a generalization of the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line as well as its clarification. The multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) are investigated. In particular, it is shown that it cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. This result generalizes the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line and clarifies it.