Abstract
Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schrödinger operator on the complete graph. The operator exhibits local quasi-modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the \({\ell^{1}}\)-though not in \({\ell^{2}}\)-sense, where the eigenvalues have the statistics of Šeba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz–Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes.
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Aizenman M., Warzel S.: Extended states in a Lifshitz tail regime for random Schrödinger operators on trees. Phys. Rev. Lett. 106, 136804 (2011)
Aizenman M., Warzel S.: Resonant delocalization for random Schrödinger operators on tree graphs. J. Eur. Math. Soc. 15, 1167–1222 (2013)
Aizenman, M., Warzel, S.: On the ubiquity of the Cauchy distribution in spectral problems (2013 preprint). arXiv:1312.7769
Albeverio S., Šeba P.: Wave chaos in quantum systems with point interactions. J. Stat. Phys. 64, 369–383 (1991)
Aleiner I.L., Matveev K.A.: Shifts of random energy levels by a local perturbation. Phys. Rev. Lett. 80, 814–817 (1998)
Berkolaiko G., Bogomolny E.B., Keating J.P.: Star graphs and Šeba billiards. J. Phys. A: Math. Gen. 34, 335–350 (2001)
Billingsley P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley-Interscience, New York (1999)
Biroli, G., Ribeiro-Teixeira, A.C., Tarzia, M.: Difference between level statistics, ergodicity and localization transitions on the Bethe lattice. arXiv:1211.7334
Bogachev L.V., Molchanov S.A.: Mean-field models in the theory of random media. I. Theor. Math. Phys. 81, 1207–1214 (1989)
Bogomolny E., Leboeuf P., Schmit C.: Spectral statistics of chaotic systems with a point like scatterer. Phys. Rev. Lett. 85, 2486–2489 (2000)
Bogomolny E., Gerland U., Schmit C.: Singular statistics. Phys. Rev. E 63, 036206 (2001)
Bogomolny E., Gerland U., Schmit C.: Nearest-neighbor distribution for singular billiards. Phys. Rev. E 65, 056214 (2002)
Bovier A.: Statistical Mechanics of Disordered Systems. Cambridge University Press, Cambridge (2006)
Altshuler A., Altshuler B.L., Kravtsov V.E., Scardicchio A.: Anderson localization on the Bethe lattice: non-ergodicity of extended states. Phys. Rev. Lett. 113, (2014)
Donoghue W.F.: The interpolation of Pick functions. Rocky Mt. J. Math. 4, 169–174 (1974)
Duren P.L.: Theory of H p spaces. Dover, New York (2000)
Dvoretzky A., Kiefer J., Wolfowitz J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 27, 642–669 (1956)
Farhi E., Goldstone J., Gutmann S., Nagaj D.: How to make the quantum adiabatic algorithm fail. Int. J. Quantum Inf. 6, 503–516 (2008)
Forrester P.J., Rains E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131, 1–61 (2005)
Keating J.P., Marklof J., Winn B.: Value distribution of the eigenfunctions and spectral determinants of quantum star graphs. Commun. Math. Phys. 241, 421–452 (2003)
Monthus C., Garel T.: Anderson localization on the Cayley tree: multifractal statistics of the transmission at criticality and off criticality. J. Phys. A 44, 145001 (2011)
Ossipov A.: Anderson localization on a simplex. J. Phys. A: Math. Theor. 46, 105001 (2013)
Šeba P.: Wave chaos in singular quantum billiard. Phys. Rev Lett. 64, 1855–1858 (1990)
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Communicated by Anton Bovier.
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Aizenman, M., Shamis, M. & Warzel, S. Resonances and Partial Delocalization on the Complete Graph. Ann. Henri Poincaré 16, 1969–2003 (2015). https://doi.org/10.1007/s00023-014-0366-9
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DOI: https://doi.org/10.1007/s00023-014-0366-9