Abstract
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.
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References
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
Band R., Berkolaiko G.: Universality of the momentum band density of periodic networks. Phys. Rev. Lett. 111, 130404 (2013)
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)
Band, R., Lévy, G.: Quantum graphs which optimize the spectral gap. Ann. Henri Poincaré 18(10), 3269–3323 (2017)
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)
Band R., Shapira T., Shapira T.: Nodal domains on isospectral quantum graphs: the resolution of isospectrality?. J. Phys. A 39(45), 13999–14014 (2006)
Barra F., Gaspard P.: On the level spacing distribution in quantum graphs. J. Stat. Phys. 101(1–2), 283–319 (2000)
Beliaev D., Kereta Z.: On the Bogomolny–Schmit conjecture. J. Phys. A 46(45), 455003, 5 (2013)
Berkolaiko G.: A lower bound for nodal count on discrete and metric graphs. Commun. Math. Phys. 278(3), 803–819 (2008)
Berkolaiko, G.: Nodal count of graph eigenfunctions via magnetic perturbation. Anal. PDE. 6,1213–1233 (2013). arXiv:1110.5373
Berkolaiko, G.: An elementary introduction to quantum graphs. In: Geometric and Computational Spectral Theory, Contemporary Mathematics, vol. 700, AMS (2017)
Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs, Volume 186 of Mathematical Surveys and Monographs. AMS, Providence (2013)
Berkolaiko, G., Latushkin, Y., Sukhtaiev, S.: On limits of quantum graph operators with shrinking edges. In preparation (2017)
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
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)
Berkolaiko G., Winn B.: Relationship between scattering matrix and spectrum of quantum graphs. Trans. Am. Math. Soc. 362(12), 6261–6277 (2010)
Blum G., Gnutzmann S., Smilansky U.: Nodal domains statistics: a criterion for quantum chaos. Phys. Rev. Lett. 88(11), 114101 (2002)
Bogomolny E., Schmit C.: Percolation model for nodal domains of chaotic wave functions. Phys. Rev. Lett. 88, 114102 (2002)
Colin de Verdière, Y.: Magnetic interpretation of the nodal defect on graphs. Anal. PDE. 6,1235–1242 (2013). Preprint. arXiv:1201.1110
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
Colin de Verdière, Y., Truc, F.: Topological resonances on quantum graphs. (2016). Preprint. arXiv:1604.01732
Courant, R.: Ein allgemeiner Satz zur Theorie der Eigenfunktione selbstadjungierter Differentialausdrücke. Nach. Ges. Wiss. Göttingen Math. Phys. Kl. 81–84 (1923)
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)
Davies EB., Pushnitski A.: Non-Weyl resonance asymptotics for quantum graphs. Anal. PDE 4, 729–756 (2011)
Diestel, R.: Graph Theory, Volume 173 of Graduate Texts in Mathematics, 4th edn. Springer, Heidelberg (2010)
Exner P., Turek, O.: Periodic quantum graphs from the Bethe–Sommerfeld perspective. J. Phys. A Math. Theor. 50(45). https://doi.org/10.1088/1751-8121/aa8d8d
Friedlander L.: Genericity of simple eigenvalues for a metric graph. Isr. J. Math. 146, 149–156 (2005)
Fulling S.A., Kuchment P., Wilson J.H.: Index theorems for quantum graphs. J. Phys. A 40(47), 14165–14180 (2007)
Gerasimenko N.I., Pavlov B.S.: A scattering problem on noncompact graphs. Teoret. Mat. Fiz. 74(3), 345–359 (1988)
Ghosh A., Reznikov A., Sarnak P.: Nodal domains of Maass forms I. Geom. Funct. Anal. 23(5), 1515–1568 (2013)
Gnutzmann S., Karageorge P.D., Smilansky U.: Can one count the shape of a drum?. Phys. Rev. Lett. 97(9), 090201, 4 (2006)
Gnutzmann S., Smilansky U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55(5–6), 527–625 (2006)
Gnutzmann S., Smilansky U., Sondergaard N.: Resolving isospectral ‘drums’ by counting nodal domains. J. Phys. A 38(41), 8921–8933 (2005)
Gnutzmann S., Smilansky U., Weber J.: Nodal counting on quantum graphs. Waves Random Media 14(1), S61–S73 (2004)
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)
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)
Karageorge PD., Smilansky U.: Counting nodal domains on surfaces of revolution. J. Phys. A 41(20), 205102 (2008)
Kennedy JB., Kurasov P., Malenová G., Mugnolo D.: On the spectral gap of a quantum graph. Ann. Henri Poincaré 17(9), 2439–2473 (2016)
Konrad, K.: Asymptotic statistics of nodal domains of quantum chaotic billiards in the semiclassical limit. Senior Thesis, Dartmouth College (2012)
Kostrykin V., Schrader R.: Kirchhoff’s rule for quantum wires. J. Phys. A 32(4), 595–630 (1999)
Kostrykin V., Schrader R.: The generalized star product and the factorization of scattering matrices on graphs. J. Math. Phys. 42(4), 1563–1598 (2001)
Kostrykin, V., Schrader, R.: Quantum wires with magnetic fluxes. Commun. Math. Phys. 237(1–2),161–179 (2003). (Dedicated to Rudolf Haag)
Kottos T., Smilansky U.: Quantum chaos on graphs. Phys. Rev. Lett. 79(24), 4794–4797 (1997)
Kottos T., Smilansky U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274(1), 76–124 (1999)
Kottos T., Smilansky U.: Chaotic scattering on graphs. Phys. Rev. Lett. 85(5), 968–971 (2000)
Mugnolo, D.: Semigroup Methods for Evolution Equations on Networks. Understanding Complex Systems. Springer, Cham (2014)
Nastasescu, M.: The number of ovals of a random real plane curve. Senior Thesis, Princeton University (2011)
Nazarov F., Sodin M.: On the number of nodal domains of random spherical harmonics. Am. J. Math. 131(5), 1337–1357 (2009)
Pleijel Åke.: Remarks on Courant’s nodal line theorem. Commun. Pure Appl. Math. 9, 543–550 (1956)
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)
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)
Schanz H., Smilansky U.: Quantization of Sinai’s billiard—a scattering approach. Chaos Solitons Fractals 5(7), 1289–1309 (1995)
Schapotschnikow P.: Eigenvalue and nodal properties on quantum graph trees. Waves Random Complex Media 16(3), 167–178 (2006)
Shmuel G., Band R.: Universality of the frequency spectrum of laminates. J. Mech. Phys. Solids 92, 127–136 (2016)
Smilansky U.: Exterior–interior duality for discrete graphs. J. Phys. A 42(3), 035101, 13 (2009)
Sturm C.: Mémoire sur les équations différentielles linéaires du second ordre. J. Math. Pures Appl. 1, 106–186 (1836)
Tutte, W.T.: Graph Theory, Volume 21 of Encyclopedia of Mathematics and Its Applications, Advanced Book Program. Addison-Wesley Publishing Company, Reading (1984)
von Below, J.: A characteristic equation associated to an eigenvalue problem on \({c^2}\)-networks. Linear Algebra Appl. 71, 309–325 (1985)
Weyl H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77(3), 313–352 (1916)
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Alon, L., Band, R. & Berkolaiko, G. Nodal Statistics on Quantum Graphs. Commun. Math. Phys. 362, 909–948 (2018). https://doi.org/10.1007/s00220-018-3111-2
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DOI: https://doi.org/10.1007/s00220-018-3111-2