To the spectral theory of vector-valued Sturm–Liouville operators with summable potentials and point interactions

Granovskyi, Yaroslav; Malamud, M. M.; Neidhardt, Hagen; Posilicano, Andrea

doi:10.4171/175-1/15

Cited by 2 publications

(5 citation statements)

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“…(ii)-(iii). Similar results for the Dirichlet realization H D is proved in [32] (see Theorem 3.4). Now the results follow from (4.34) by applying the Kato-Rosenblum and Weyl theorems on the stability of the absolutely continuous and continuous spectra of H D respectively, under trace class perturbations (see [1], [53], [57]).…”

Section: 15)supporting

confidence: 78%

“…In particular, the Hamiltonians H α,Q have no sc-spectrum. Theorem 1.1 generalizes the main result of the paper [32] to the case of quantum graphs under discussion. A special case of Theorem 1.1 with Q = 0 was established by B.-S. Ong [49] by using the limit absorption principle.…”

Section: Introductionsupporting

confidence: 65%

“…. = α N = 0 each realization in L 2 (R + ; C m ) of Sturm-Liouville expression with a potential matrix Q ∈ L 1 (R + ; C m×m ) has the purely absolutely continuous spectrum of the multiplicity m (see [32]). In turn, this result generalizes the classical Titchmarsh's result for the scalar operator associated in L 2 (R + ) with the expression A := − d 2 dx 2 + q, where q = q ∈ L 1 (R + ) (see Chapter 5 in [55]).…”

Section: 38)mentioning

confidence: 99%

“…We also present necessary facts on description of absolutely continuous and singular continuous spectra of extensions in terms of the limit behavior of the Weyl function on the real axis. In Section 3 following [32] we present the necessary information on the spectral theory of Schrödinger operators on L 2 (R; C m ) with summable potential matrix Q = Q * .…”

Section: Introductionmentioning

confidence: 99%

“…(i) Proposition 4.7 is also valid for realizations of A X,α,Q in L 2 (R + ; C m ) including the Hamiltonian H X,α,Q (cf [32,. Theorem 4.6]);(ii) In the case α 1 = .…”

mentioning

confidence: 99%

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Non-compact Quantum Graphs with Summable Matrix Potentials

Granovskyi

Malamud

Neidhardt

2020

Ann. Henri Poincaré

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Let G be a metric noncompact connected graph with finitely many edges. The main object of the paper is the Hamiltonian H α associated in L 2 (G; C m) with a matrix Sturm-Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian H α as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of H α. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of H α is obtained. Additionally, for a star graph G a formula is found for the scattering matrix of the pair {H α , H D }, where H D is the Dirichlet operator on G.

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Section: 15)supporting

confidence: 78%

Section: Introductionsupporting

confidence: 65%

Section: 38)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…(i) Proposition 4.7 is also valid for realizations of A X,α,Q in L 2 (R + ; C m ) including the Hamiltonian H X,α,Q (cf [32,. Theorem 4.6]);(ii) In the case α 1 = .…”

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confidence: 99%

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