Nodal Statistics on Quantum Graphs


It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the “nodal surplus”) for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graph’s first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form—it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.

This is a preview of subscription content, access via your institution.


  1. 1

    Band, R.: The nodal count \({\{0,1,2,3,\ldots\}}\) implies the graph is a tree. Philos. Trans. R. Soc. Lond. A. 372(2007),20120504, 24 (2014). arXiv:1212.6710

  2. 2

    Band R., Berkolaiko G.: Universality of the momentum band density of periodic networks. Phys. Rev. Lett. 111, 130404 (2013)

    ADS  Article  Google Scholar 

  3. 3

    Band R., Berkolaiko G., Smilansky U.: Dynamics of nodal points and the nodal count on a family of quantum graphs. Ann. Henri Poincare 13(1), 145–184 (2012)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Band, R., Lévy, G.: Quantum graphs which optimize the spectral gap. Ann. Henri Poincaré 18(10), 3269–3323 (2017)

  5. 5

    Band, R., Oren, I., Smilansky, U.: Nodal domains on graphs—how to count them and why? In: Analysis on Graphs and Its Applications, Volume 77 of Proceedings of Symposia in Pure Mathematics, pp. 5–27. American Mathematical Society, Providence, RI (2008)

  6. 6

    Band R., Shapira T., Shapira T.: Nodal domains on isospectral quantum graphs: the resolution of isospectrality?. J. Phys. A 39(45), 13999–14014 (2006)

  7. 7

    Barra F., Gaspard P.: On the level spacing distribution in quantum graphs. J. Stat. Phys. 101(1–2), 283–319 (2000)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Beliaev D., Kereta Z.: On the Bogomolny–Schmit conjecture. J. Phys. A 46(45), 455003, 5 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Berkolaiko G.: A lower bound for nodal count on discrete and metric graphs. Commun. Math. Phys. 278(3), 803–819 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Berkolaiko, G.: Nodal count of graph eigenfunctions via magnetic perturbation. Anal. PDE. 6,1213–1233 (2013). arXiv:1110.5373

  11. 11

    Berkolaiko, G.: An elementary introduction to quantum graphs. In: Geometric and Computational Spectral Theory, Contemporary Mathematics, vol. 700, AMS (2017)

  12. 12

    Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs, Volume 186 of Mathematical Surveys and Monographs. AMS, Providence (2013)

  13. 13

    Berkolaiko, G., Latushkin, Y., Sukhtaiev, S.: On limits of quantum graph operators with shrinking edges. In preparation (2017)

  14. 14

    Berkolaiko, G., Liu, W.: Simplicity of eigenvalues and non-vanishing of eigenfunctions of a quantum graph. J. Math. Anal. Appl. 445(1), 803–818 (2017). arXiv:1601.06225

  15. 15

    Berkolaiko G., Weyand T.: Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2007), 20120522, 17 (2014)

    MathSciNet  MATH  Google Scholar 

  16. 16

    Berkolaiko G., Winn B.: Relationship between scattering matrix and spectrum of quantum graphs. Trans. Am. Math. Soc. 362(12), 6261–6277 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Blum G., Gnutzmann S., Smilansky U.: Nodal domains statistics: a criterion for quantum chaos. Phys. Rev. Lett. 88(11), 114101 (2002)

    ADS  Article  Google Scholar 

  18. 18

    Bogomolny E., Schmit C.: Percolation model for nodal domains of chaotic wave functions. Phys. Rev. Lett. 88, 114102 (2002)

    ADS  Article  Google Scholar 

  19. 19

    Colin de Verdière, Y.: Magnetic interpretation of the nodal defect on graphs. Anal. PDE. 6,1235–1242 (2013). Preprint. arXiv:1201.1110

  20. 20

    Colin de Verdière, Y.: Semi-classical measures on quantum graphs and the Gauß map of the determinant manifold. Ann. Henri Poincaré 16(2),347–364 (2015). arXiv:1311.5449

  21. 21

    Colin de Verdière, Y., Truc, F.: Topological resonances on quantum graphs. (2016). Preprint. arXiv:1604.01732

  22. 22

    Courant, R.: Ein allgemeiner Satz zur Theorie der Eigenfunktione selbstadjungierter Differentialausdrücke. Nach. Ges. Wiss. Göttingen Math. Phys. Kl. 81–84 (1923)

  23. 23

    Davies E.B., Exner P., Lipovský J.: Non-Weyl asymptotics for quantum graphs with general coupling conditions. J. Phys. A 43(47), 474013, 16 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Davies EB., Pushnitski A.: Non-Weyl resonance asymptotics for quantum graphs. Anal. PDE 4, 729–756 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25

    Diestel, R.: Graph Theory, Volume 173 of Graduate Texts in Mathematics, 4th edn. Springer, Heidelberg (2010)

  26. 26

    Exner P., Turek, O.: Periodic quantum graphs from the Bethe–Sommerfeld perspective. J. Phys. A Math. Theor. 50(45).

  27. 27

    Friedlander L.: Genericity of simple eigenvalues for a metric graph. Isr. J. Math. 146, 149–156 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28

    Fulling S.A., Kuchment P., Wilson J.H.: Index theorems for quantum graphs. J. Phys. A 40(47), 14165–14180 (2007)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  29. 29

    Gerasimenko N.I., Pavlov B.S.: A scattering problem on noncompact graphs. Teoret. Mat. Fiz. 74(3), 345–359 (1988)

    MathSciNet  Google Scholar 

  30. 30

    Ghosh A., Reznikov A., Sarnak P.: Nodal domains of Maass forms I. Geom. Funct. Anal. 23(5), 1515–1568 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31

    Gnutzmann S., Karageorge P.D., Smilansky U.: Can one count the shape of a drum?. Phys. Rev. Lett. 97(9), 090201, 4 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. 32

    Gnutzmann S., Smilansky U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55(5–6), 527–625 (2006)

    ADS  Article  Google Scholar 

  33. 33

    Gnutzmann S., Smilansky U., Sondergaard N.: Resolving isospectral ‘drums’ by counting nodal domains. J. Phys. A 38(41), 8921–8933 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  34. 34

    Gnutzmann S., Smilansky U., Weber J.: Nodal counting on quantum graphs. Waves Random Media 14(1), S61–S73 (2004)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  35. 35

    Jung J., Zelditch S.: Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution. J. Differ. Geom. 102(1), 37–66 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36

    Jung J., Zelditch S.: Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary. Math. Ann. 364(3-4), 813–840 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37

    Karageorge PD., Smilansky U.: Counting nodal domains on surfaces of revolution. J. Phys. A 41(20), 205102 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  38. 38

    Kennedy JB., Kurasov P., Malenová G., Mugnolo D.: On the spectral gap of a quantum graph. Ann. Henri Poincaré 17(9), 2439–2473 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. 39

    Konrad, K.: Asymptotic statistics of nodal domains of quantum chaotic billiards in the semiclassical limit. Senior Thesis, Dartmouth College (2012)

  40. 40

    Kostrykin V., Schrader R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32(4), 595–630 (1999)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. 41

    Kostrykin V., Schrader R.: The generalized star product and the factorization of scattering matrices on graphs. J. Math. Phys. 42(4), 1563–1598 (2001)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  42. 42

    Kostrykin, V., Schrader, R.: Quantum wires with magnetic fluxes. Commun. Math. Phys. 237(1–2),161–179 (2003). (Dedicated to Rudolf Haag)

  43. 43

    Kottos T., Smilansky U.: Quantum chaos on graphs. Phys. Rev. Lett. 79(24), 4794–4797 (1997)

    ADS  Article  Google Scholar 

  44. 44

    Kottos T., Smilansky U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274(1), 76–124 (1999)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  45. 45

    Kottos T., Smilansky U.: Chaotic scattering on graphs. Phys. Rev. Lett. 85(5), 968–971 (2000)

    ADS  Article  Google Scholar 

  46. 46

    Mugnolo, D.: Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems. Springer, Cham (2014)

  47. 47

    Nastasescu, M.: The number of ovals of a random real plane curve. Senior Thesis, Princeton University (2011)

  48. 48

    Nazarov F., Sodin M.: On the number of nodal domains of random spherical harmonics. Am. J. Math. 131(5), 1337–1357 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49

    Pleijel Åke.: Remarks on Courant’s nodal line theorem. Commun. Pure Appl. Math. 9, 543–550 (1956)

    MathSciNet  Article  MATH  Google Scholar 

  50. 50

    Pokornyĭ Yu V., Pryadiev VL., Al’-Obeĭd A.: On the oscillation of the spectrum of a boundary value problem on a graph. Mat. Zametki 60(3), 468–470 (1996)

    MathSciNet  Article  Google Scholar 

  51. 51

    Rouvinez C., Smilansky U.: A scattering approach to the quantization of Hamiltonians in two dimensions—application to the wedge billiard. J. Phys. A 28(1), 77–104 (1995)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  52. 52

    Schanz H., Smilansky U.: Quantization of Sinai’s billiard—a scattering approach. Chaos Solitons Fractals 5(7), 1289–1309 (1995)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  53. 53

    Schapotschnikow P.: Eigenvalue and nodal properties on quantum graph trees. Waves Random Complex Media 16(3), 167–178 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  54. 54

    Shmuel G., Band R.: Universality of the frequency spectrum of laminates. J. Mech. Phys. Solids 92, 127–136 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  55. 55

    Smilansky U.: Exterior–interior duality for discrete graphs. J. Phys. A 42(3), 035101, 13 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  56. 56

    Sturm C.: Mémoire sur les équations différentielles linéaires du second ordre. J. Math. Pures Appl. 1, 106–186 (1836)

    Google Scholar 

  57. 57

    Tutte, W.T.: Graph Theory, Volume 21 of Encyclopedia of Mathematics and Its Applications, Advanced Book Program. Addison-Wesley Publishing Company, Reading (1984)

  58. 58

    von Below, J.: A characteristic equation associated to an eigenvalue problem on \({c^2}\)-networks. Linear Algebra Appl. 71, 309–325 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  59. 59

    Weyl H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77(3), 313–352 (1916)

    Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Gregory Berkolaiko.

Additional information

Communicated by J. Marklof

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alon, L., Band, R. & Berkolaiko, G. Nodal Statistics on Quantum Graphs. Commun. Math. Phys. 362, 909–948 (2018).

Download citation