Advertisement

Annales Henri Poincaré

, Volume 20, Issue 5, pp 1517–1542 | Cite as

Optimal Potentials for Quantum Graphs

  • Pavel KurasovEmail author
  • Andrea Serio
Open Access
Article
  • 112 Downloads

Abstract

Schrödinger operators on metric graphs with delta couplings at the vertices are studied. We discuss which potential and which distribution of delta couplings on a given graph maximise the ground state energy, provided the integral of the potential and the sum of strengths of the delta couplings are fixed. It appears that the optimal potential if it exists is a constant function on its support formed by a set of intervals separated from the vertices. In the case where the optimal configuration does not exist explicit optimising sequences are presented.

Mathematics Subject Classification

35P15 81Q35 

Notes

Acknowledgements

We wish to thank Gregory Berkolaiko, James Kennedy and Delio Mugnolo for the discussion regarding the application of the cutting principles. The authors were partially supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld in the framework of the cooperation group Discrete and continuous models in the theory of networks. P. K. was partially supported by the Swedish Research Council Grant D0497301. We would like to thank anonymous referees for careful reading of the manuscript and constructive suggestions.

References

  1. 1.
    Albeverio, S., Kurasov, P.: Singular perturbations of differential operators, London Mathematical Society Lecture Note Series, vol. 271. Cambridge University Press, Cambridge (2000). Solvable Schrödinger type operatorsGoogle Scholar
  2. 2.
    Band, R., Lévy, G.: Quantum graphs which optimize the spectral gap. Ann. Henri Poincaré 18(10), 3269–3323 (2017).  https://doi.org/10.1007/s00023-017-0601-2 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berkolaiko, G., Kennedy, J.B., Kurasov, P., Mugnolo, D.: Edge connectivity and the spectral gap of combinatorial and quantum graphs. J. Phys. A 50(36), 365201 (2017).  https://doi.org/10.1088/1751-8121/aa8125 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berkolaiko, G., Kennedy, J., Kurasov, P., Mugnolo, D.: Surgery principles for the spectral analysis of quantum graphs (2018). arXiv:1807.08183
  5. 5.
    Berkolaiko, G.: Introduction to Quantum Graphs. Mathematical Surveys and Monographs, vol. 186. American Mathematical Society, Providence (2013)zbMATHGoogle Scholar
  6. 6.
    Boman, J., Kurasov, P., Suhr, R.: Schrödinger operators on graphs and geometry II. Spectral estimates for \(L_{1}\)-potentials and an Ambartsumian theorem. Integr. Equ. Oper. Theory 90, 40 (2018).  https://doi.org/10.1007/s00020-018-2467-1 CrossRefzbMATHGoogle Scholar
  7. 7.
    Egnell, H.: Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14(1), 1–48 (1987)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Essén, M.: Optimization and rearrangements of the coefficient in the differential equation(s) \(y^{\prime \prime }\pm qy=0\), C. R. Math. Rep. Acad. Sci. Can. 6(1), 15–20 (1984)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Essén, M.: On estimating Eigenvalues of a second order linear differential operator. General Inequal., 5 (Oberwolfach, 1986). Int. Schriftenreihe Numer. Math., vol. 80. Birkhäuser, Basel, pp. 347–366 (1987)Google Scholar
  10. 10.
    Ezhak, S.S.: On estimates for the minimum eigenvalue of the Sturm-Liouville problem with an integrable condition, Sovrem. Mat. Prilozh. 36, 56–69 (2005).  https://doi.org/10.1007/s10958-007-0345-5 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 145(5), 5205–5218 (2007)
  11. 11.
    Harrell, I.I., Evans, M.: Hamiltonian operators with maximal eigenvalues. J. Math. Phys. 25(1), 48–51 (1984).  https://doi.org/10.1063/1.525996 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Karreskog, G., Kurasov, P., Trygg Kupersmidt, I.: Schrödinger operators on graphs: symmetrization and Eulerian cycles. Proc. Am. Math. Soc. 144(3), 1197–1207 (2016).  https://doi.org/10.1090/proc12784 CrossRefzbMATHGoogle Scholar
  13. 13.
    Karulina, E.S., Vladimirov, A.A.: The Sturm-Liouville problem with singular potential and the extrema of the first eigenvalue. Tatra Mt. Math. Publ. 54, 101–118 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kennedy, J.B., Kurasov, P., Malenová, G., Mugnolo, D.: On the spectral gap of a quantum graph. Ann. HenriPoincaré 17(9), 2439–2473 (2016).  https://doi.org/10.1007/s00023-016-0460-2 ADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    Kurasov, P.: Schrödinger operators on graphs and geometry. I. Essentially bounded potentials. J. Funct. Anal. 254(4), 934–953 (2008).  https://doi.org/10.1016/j.jfa.2007.11.007 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kurasov, P.: Quantum graphs: spectral theory and inverse problems, to appear in BirkhäauserGoogle Scholar
  17. 17.
    Kurasov, P.: On the ground state for Quantum Graphs. Research Reports in Mathematics, no. 1, Dept. of Mathematics, Stockholm Univ. (submitted) (2019)Google Scholar
  18. 18.
    Kurasov, P., Malenová, G., Naboko, S.: Spectral gap for quantum graphs and their edge connectivity. J. Phys. A 46(27), 275309 (2013).  https://doi.org/10.1088/1751-8113/46/27/275309 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kurasov, P., Naboko, S.: Rayleigh estimates for differential operators on graphs. J. Spectr. Theory 4(2), 211–219 (2014).  https://doi.org/10.4171/JST/67 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Post, O.: Spectral Analysis on Graph-Like Spaces. Lecture Notes in Mathematics, vol. 2039. Springer, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  21. 21.
    Ramm, A.G.: Problem list in queries. Notices Am. Math. Soc. 29, 326–329 (1982)Google Scholar
  22. 22.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional analysis. Academic Press, New York, London (1972)zbMATHGoogle Scholar
  23. 23.
    Talenti, G.: Estimates for Eigenvalues of Sturm-Liouville problems. General Inequal., 4 (Oberwolfach, 1983). Int. Schriftenreihe Numer. Math., vol. 71. Birkhäauser, Basel, pp. 341–350 (1984)Google Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden

Personalised recommendations