Advertisement

Doklady Mathematics

, Volume 100, Issue 2, pp 405–410 | Cite as

Quantum Graphs with Summable Matrix Potentials

  • Ya. I. GranovskyiEmail author
  • M. M. MalamudEmail author
  • H. Neidhardt
MATHEMATICS
  • 7 Downloads

Abstract

Let \(\mathcal{G}\) be a metric, finite, noncompact, and connected graph with finitely many edges and vertices. Assume that the length of at least one of the edges is infinite. The main object of this 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 \({{{\mathbf{H}}}_{\alpha }}\) and any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring that the positive part of \({\mathbf{H}_{\alpha }}\) is purely absolutely continuous. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of the operator \({{{\mathbf{H}}}_{\alpha }}\) is obtained. Additionally, a formula is found for the scattering matrix of the pair \(\{ {{{\mathbf{H}}}_{\alpha }},{{{\mathbf{H}}}_{D}}\} \), where \({{{\mathbf{H}}}_{D}}\) is the operator of the Dirichlet problem on the graph.

Notes

FUNDING

This work was supported by the RUDN 5-100 program.

REFERENCES

  1. 1.
    J. Behrndt, M. M. Malamud, and H. Neidhardt, J. Funct. Anal. 273, 1970–2025 (2017).MathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs (Am. Math. Soc. Providence, R.I., 2013).zbMATHGoogle Scholar
  3. 3.
    N. I. Gerasimenko and B. S. Pavlov, TMF 74 (3), 345–359 (1988).Google Scholar
  4. 4.
    E. Davies and A. Pushnitski, J. Anal. PDE 4 (5), (2011).Google Scholar
  5. 5.
    V. A. Derkach and M. M. Malamud, Theory of Operator Extensions and Boundary Value Problems (Inst. Mat. Nats. Akad. Nauk Ukr., Kiev, 2017).Google Scholar
  6. 6.
    V. A. Derkach and M. M. Malamud, J. Funct. Anal. 95, 1–95 (1991).MathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Exner, A. Kostenko, M. Malamud, and H. Neidhard, Ann. Henri Poincare 19 (11), 3457–3510 (2017).CrossRefGoogle Scholar
  8. 8.
    P. Exner, A. Laptev, and M. Usman, Commun. Math. Phys. 326, 531–541 (2014).CrossRefGoogle Scholar
  9. 9.
    Ya. Granovskyi, M. Malamud, H. Neidhardt, and A. Posilicano, in Functional Analysis and Operator Theory for Quantum Physics: The Pavel Exner Anniversary Volume (Eur. Math. Soc., Zurich, 2017), pp. 271–313.Google Scholar
  10. 10.
    M. M. Malamud and H. Neidhard, J. Funct. Anal. 260 (3), 613–638 (2011).MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. P. Maslov, Operational Methods (Nauka, Moscow, 1973; Mir, Moscow, 1976).Google Scholar
  12. 12.
    B.-S. Ong, in Quantum Graphs and Their Applications (Am. Math. Soc. Providence, R.I., 2006), pp. 241–249.Google Scholar
  13. 13.
    O. Post, Lecture Notes in Mathematics (Springer, Berlin, 2012), Vol. 2039.Google Scholar
  14. 14.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, 2nd ed. (Academic, New York, 1980).zbMATHGoogle Scholar
  15. 15.
    E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations (Clarendon, Oxford, 1946).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Mechanics DonetskUkraine
  2. 2.RUDN UniversityMoscowRussia

Personalised recommendations