Abstract
The subject of this work is random Schrödinger operators on regular rooted tree graphs with stochastically homogeneous disorder. The operators are of the form H λ (ω)=T+U+λ V(ω) acting in ℓ2(), with T the adjacency matrix, U a radially periodic potential, and V(ω) a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of H λ (ω) exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder in case U=0 and V(ω) given by iid random variables on . Our main contribution is a new method for establishing the persistence of ac spectrum under weak disorder. The method yields the continuity of the ac spectral density of H λ (ω) at λ=0. The latter is shown to converge in the L 1-sense over closed Borel sets in which H 0 has no singular spectrum. The analysis extends to random potentials whose values at different sites need not be independent, assuming only that their joint distribution is weakly correlated across different tree branches.
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Abou-Chacra, R., Anderson, P.W., Thouless, D.J.: A selfconsistent theory of localization. J. Phys. C: Solid State Phys. 6, 1734–1752 (1973)
Abou-Chacra, R., Thouless, D.J.: Self-consistent theory of localization. II. localization near the band edges. J. Phys. C: Solid State Phys. 7, 65–75 (1974)
Aizenman, M., Molchanov, S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245 (1993)
Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6, 1163–1182 (1994)
Aizenman, M., Warzel, S.: Persistence under weak disorder of AC spectra of quasi-periodic Schrödinger operators on tree graphs. Preprint math-ph/0504084. To appear in Mosc. Math. J.
Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)
Bauer, H.: Measure and integration theory. de Gruyter, Berlin, 2001
Billingsley, P.: Convergence of probability measures. Wiley, New York, 1968
Carmona, R., Lacroix, J.: Spectral theory of random Schrödinger operators. Birkhäuser, Boston, 1990
Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Theor. Prob. Appl. 13, 197–224 (1968)
Duren, P.L.: Theory of H p spaces. Academic, New York, 1970
Froese, R., Hasler, D., Spitzer, W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230, 184–221 (2006)
Georgii, H.-O.: Gibbs measures and phase transitions. de Gruyter, Berlin, 1988
Goldsheid, I.Ya., Molchanov, S., Pastur, L.: A pure point spectrum of the stochastic one-dimensional schrödinger operator. Funct. Anal. Appl. 11, 1–8 (1977)
Hupfer, T., Leschke, H., Müller, P., Warzel, S.: Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials. Rev. Math. Phys. 13, 1547–1581 (2001)
Ishii, K.: Localization of eigenstates and transport phenomena in the one-dimensional disordered system. Supp. Progr. Theor. Phys. 53, 77–138 (1973)
Jitormirskaya, S.Ya.: Metal-insulator transition for the almost-Mathieu operator. Ann. Math. 150, 1159–1175 (1999)
Klein, A.: The Anderson metal-insulator transition on the Bethe lattice. In: Iagolnitzer, D (ed) Proceedings of the XIth international congress on mathematical physics, Paris, France, July 18–23, 1994, pp. 383–391. International Press, Cambridge, MA, 1995
Klein, A.: Spreading of wave packets in the Anderson model on the Bethe lattice. Commun. Math. Phys. 177, 755–773 (1996)
Klein, A.: Extended states in the Anderson model on the Bethe lattice. Adv. Math. 133, 163–184 (1998)
Kotani, S.: Ljapunov indices determine absolute continuous spectra of stationary one dimensional Schrödinger operators. In: Ito, K (ed) Proc. Taneguchi Itern. Symp. on Stochastic Ananlysis, Amsterdam, North Holland, 1983, pp. 225–247
Kotani, S.: One-dimensional random Schrödinger operators and Herglotz functions. In: Ito, K (ed) Taneguchi Symp. PMMP, Amsterdam, North Holland, 1985, pp. 219–250
Lieb ans, E.H., Loss, M.: Analysis, 2 nd edn., Amer. Math. Soc. Providence, RI, 2001
Miller, J.D., Derrida, B.: Weak disorder expansion for the Anderson model on a tree. J. Stat. Phys. 75, 357–388 (1993)
Mirlin, A.D., Fyodorov, Y.V.: Localization transition in the Anderson model on the Bethe lattice: spontaneous symmetry breaking and correlation functions. Nucl. Phys. B. 366, 507–532 (1991)
Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. Springer-Verlag, Berlin, 1992
Pastur, L.A.: Spectral properties of disordered systems in the one body approximation. Commun. Math. Phys. 75, 167–196 (1980)
Reed, M., Simon, B.: Methods of modern mathematical physics I: Functional analysis, 2nd edn., Academic Press Inc., New York, 1980
Rudin W.: Real and complex analysis, 3rd edn., McGraw-Hill, New York, 1987
Simon, B.: Kotani theory for one-dimensional Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983)
Stollmann, P.: Caught by disorder: bound states in random media. Birkhäuser, Boston, 2001
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Michael Aizenman: Visiting the Department of Physics of Complex System, Weizmann Inst. of Science, Israel
Simone Warzel: On leave from: Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Germany
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Aizenman, M., Sims, R. & Warzel, S. Stability of the Absolutely Continuous Spectrum of Random Schrödinger Operators on Tree Graphs. Probab. Theory Relat. Fields 136, 363–394 (2006). https://doi.org/10.1007/s00440-005-0486-8
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DOI: https://doi.org/10.1007/s00440-005-0486-8