Abstract
We compute the value distributions of the eigenfunctions and spectral determinant of the Schrödinger operator on families of star graphs. The values of the spectral determinant are shown to have a Cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related. This is in contrast to the spectral determinants of random matrices, for which the logarithm is known to satisfy a Gaussian limit distribution. The value distribution of the eigenfunctions also differs from the corresponding random matrix result. We argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of Šeba-type billiards.
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Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. New York: Dover Publishing, 1965
Barra, F., Gaspard, P.: On the level spacing distribution in quantum graphs. J. Stat. Phys. 101, 283–319 (2000)
Berkolaiko, G., Bogomolny, E.B., Keating, J.P.: Star graphs and Šeba billiards. J. Phys. A 34, 335–350 (2001)
Berkolaiko, G., Keating, J.P.: Two-point spectral correlations for star graphs. J. Phys. A 32, 7827–7841 (1999)
Berkolaiko, G., Schanz, H., Whitney, R.S.: Leading off-diagonal correction to the form factor of large graphs. Phys. Rev. Lett. 88, art. no. 104101 (2002)
Berry, M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A 10, 2083–2091 (1977)
Berry, M.V., Tabor, M.: Level clustering in the regular spectrum. Proc. Roy. Soc. Lond. A 356, 375–394 (1977)
Bogomolny, E.B., Gerland, U., Schmit, C.: Models of intermediate spectral statistics. Phys. Rev. E 59, R1315–R1318 (1999)
Bogomolny, E.B., Gerland, U., Schmit, C.: Singular statistics. Phys. Rev. E 63, art. no. 036206 (2001)
Bogomolny, E.B., Leboeuf, P., Schmit, C.: Spectral statistics of chaotic systems with a pointlike scatterer. Phys. Rev. Lett. 85, 2486–2489 (2000)
Bolte, J., Harrison, J.: Spectral statistics for the Dirac operator on graphs. J. Phys. A 36, 2747–2769 (2003)
Desbois, J.: Spectral determinant of Schrödinger operator on graphs. J. Phys. A 33, 63–67 (2000)
Eskin, A., Margulis, G., Mozes, S.: Quadratic forms of signature (2,2) and eigenvalue spacings on rectangular 2-tori. Preprint available at www.math.uchicago.edu/~eskin/, 2001
Feller, W.: An introduction to probability theory and its applications. New York: Wiley, 1971
Haake, F., Życzkowski, K.: Random-matrix theory and eigenmodes of dynamical systems. Phys. Rev. A 42, 1013–1016 (1990)
Hughes, C.P., Keating, J.P., O’Connell, N.: Random matrix theory and the derivative of the Riemann zeta function. Proc. Roy. Soc: A 456, 2611–2627 (2000)
Keating, J.P., Snaith, N.C.: Random matrix theory and ζ(1/2+it). Commun. Math. Phys. 214, 57–89 (2000)
Kottos, T., Schanz, H.: Quantum graphs: a model for quantum chaos. Physica E 9, 523–530 (2001)
Kottos, T., Smilansky, U.: Quantum Chaos on graphs Phys. Rev. Lett. 79, 4794–4797 (1997)
Kottos, T., Smilansky, U.: Periodic orbit theory and spectral statistics for quantum graphs. Ann. Phys. 274, 76–124 (1999)
Kurasov, P., Stenberg, F.: On the inverse scattering problem on branching graphs. J. Phys. A 35, 101–121 (2002)
Kurlberg, P., Rudnick, Z.: Value distribution for eigenfunctions of desymmetrized quantum maps. Int. Math. Res. Notices 18, 985–1002 (2001)
Marklof, J.: Spectral form factors of rectangle billiards. Commun. Math. Phys. 199, 169–202 (1998)
Pakoński, P., Życzkowski, K., Kuś, M.: Classical 1D maps, quantum graphs and ensembles of unitary matrices. J. Phys. A 34, 9303–9319 (2001)
Pascaud, M., Montambaux, G.: Persistent currents on networks. Phys. Rev. Lett. 82, 4512–4515 (1999)
Schanz, H., Smilansky, U.: Spectral statistics for quantum graphs: periodic orbits and combinatorics. In: Proceedings of the Australian summer school on quantum chaos and mesoscopics, Canberra, 1999
Schanz, H., Smilansky, U.: Periodic-orbit theory of Anderson localisation on graphs. Phys. Rev. Lett. 84, 1472–1430 (2000)
Šeba, P.: Wave chaos in singular quantum billiard. Phys. Rev. Lett. 64, 1855–1858 (1990)
Šeba, P., Albeverio, S.: Wave chaos in quantum systems with point interaction. J. Stat. Phys 64, 369–383 (1991)
Tanner, G.: Unitary stochastic matrix ensembles and spectral statistics. J. Phys. A 34, 8485–8500 (2001)
Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916)
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Keating, J., Marklof, J. & Winn, B. Value Distribution of the Eigenfunctions and Spectral Determinants of Quantum Star Graphs. Commun. Math. Phys. 241, 421–452 (2003). https://doi.org/10.1007/s00220-003-0941-2
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DOI: https://doi.org/10.1007/s00220-003-0941-2