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Non-compact Quantum Graphs with Summable Matrix Potentials

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Abstract

Let \(\mathcal {G}\) be a metric non-compact connected graph with finitely many edges. The main object of the paper is the Hamiltonian \(\mathbf{H}_{\alpha }\) associated in \(L^2(\mathcal {G};\mathbb {C}^m)\) 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 \(\mathbf{H}_{\alpha }\) 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 \(\mathbf{H}_{\alpha }\). Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of \(\mathbf{H}_{\alpha }\) is obtained. Additionally, for a star graph \(\mathcal {G}\) a formula is found for the scattering matrix of the pair \(\{\mathbf{H}_{\alpha }, \mathbf{H}_D\}\), where \(\mathbf{H}_D\) is the Dirichlet operator on \(\mathcal {G}\).

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Acknowledgements

The authors are indebted to anonymous referees for useful remarks and comments allowing us to improve the exposition.

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Author notes

  1. Hagen Neidhardt deceased on 23 March 2019.

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, R. Luxemburg Str. 74, Donetsk, Ukraine

    Yaroslav Granovskyi

  2. Peoples Friendship University of Russia (RUDN University), Miklukho-Maklaya Str. 6, Moscow, Russian Federation, 117198

    Mark Malamud

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117, Berlin, Germany

    Hagen Neidhardt

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  1. Yaroslav Granovskyi
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  2. Mark Malamud
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  3. Hagen Neidhardt
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Correspondence to Mark Malamud.

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Communicated by Jan Derezinski.

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Research supported by the “RUDN University Program 5-100” (M.M.) and by the European Research Council (ERC) under Grant No. AdG 267802 “AnaMultiScale” (H.N.).

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Granovskyi, Y., Malamud, M. & Neidhardt, H. Non-compact Quantum Graphs with Summable Matrix Potentials. Ann. Henri Poincaré 22, 1–47 (2021). https://doi.org/10.1007/s00023-020-00977-3

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