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Localization on quantum graphs with random vertex couplings. (English) Zbl 1144.82061
Summary: We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. We obtain necessary conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges.

MSC:
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
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[1] Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6, 1163–1182 (1994). Special issue · Zbl 0843.47039
[2] Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. 224, 219–253 (2001) · Zbl 1038.82038
[3] Aizenman, M., Elgart, A., Naboko, S., Schenker, J.H., Stolz, G.: Moment analysis of localization in random Schroedinger operators. Invent. Math. 163, 343–413 (2006) · Zbl 1090.81026
[4] Aizenman, M., Sims, R., Warzel, S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun. Math. Phys. 264, 371–389 (2006) · Zbl 1233.34009
[5] Amrein, W.O., Georgescu, V.: On the characterization of bound states and scattering states in quantum mechanics. Helvetica Phys. Acta 46, 635–658 (1973)
[6] Boutet de Monvel, A., Grinshpun, V.: Exponential localization for multidimensional Schrödinger operators with random point potentials. Rev. Math. Phys. 9, 425–451 (1997) · Zbl 0895.60103
[7] Brüning, J., Geyler, V., Pankrashkin, K.: Cantor and band spectra for periodic quantum graphs with magnetic fields. Commun. Math. Phys. 269, 87–105 (2007) · Zbl 1113.81053
[8] Brüning, J., Geyler, V., Pankrashkin, K.: Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. Rev. Math. Phys. 20, 1–70 (2008). Preprint arXiv:math-ph/0611088 · Zbl 1163.81007
[9] Dorlas, T.C., Macris, N., Pulé, J.V.: Characterization of the spectrum of the Landau Hamiltonian with delta impurities. Commun. Math. Phys. 204, 367–396 (1999) · Zbl 0937.60063
[10] Exner, P.: Lattice Kronig-Penney models. Phys. Rev. Lett. 74, 3503–3506 (1995)
[11] Exner, P., Helm, M., Stollmann, P.: Localization on a quantum graph with a random potential on the edges. Rev. Math. Phys. 19, 923–939 (2007) · Zbl 1147.82018
[12] Geyler, V.A., Margulis, V.A.: Anderson localization in the nondiscrete Maryland model. Theor. Math. Phys. 70, 133–140 (1987)
[13] Gnutzmann, S., Smilansky, U.: Quantum graphs: applications to quantum chaos and universal spectral statistics. Adv. Phys. 55, 527–625 (2006)
[14] Gruber, M.J., Lenz, D., Veselić, I.: Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over \(\mathbb{Z}\) d . J. Funct. Anal. 253, 515–533 (2007). Preprint arXiv:math.SP/0612743 · Zbl 1133.35030
[15] Helm, M., Veselić, I.: A linear Wegner estimate for alloy type Schrödinger operators on metric graphs. J. Math. Phys. 48, 092107 (2007) · Zbl 1152.81470
[16] Hislop, P.D., Post, O.: Anderson localization for radial tree-like random quantum graphs. Preprint arXiv:math-ph/0611022
[17] Hislop, P.D., Kirsch, W., Krishna, M.: Spectral and dynamical properties of random models with non-local and singular interactions. Math. Nachr. 278, 627–664 (2005) · Zbl 1123.82011
[18] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966) · Zbl 0148.12601
[19] Klopp, F.: Localization for semi-classical continuous random Schrödinger operators II: the random displacement model. Helvetica Phys. Acta 66, 810–841 (1993) · Zbl 0820.60043
[20] Klopp, F.: Localisation pour des opérateurs de Schrödinger aléatoires dans L 2(R d ): un modèle semi-classique. Ann. Inst. Fourier 45, 265–316 (1995) · Zbl 0817.35088
[21] Klopp, F.: Band edge behaviour for the integrated density of states of random Jacobi matrices in dimension 1. J. Stat. Phys. 90, 927–947 (1998) · Zbl 0924.47057
[22] Klopp, F.: Weak disorder localization and Lifshitz tails. Commun. Math. Phys. 232, 125–155 (2002) · Zbl 1034.82024
[23] Klopp, F., Wolff, T.: Lifschitz tails for 2-dimensional random Schrödinger operators. J. Anal. Math. 88, 63–147 (2002) · Zbl 1058.47034
[24] Kostrykin, V., Schrader, R.: A random necklace model. Waves Random Media 14, S75–S90 (2004) · Zbl 1063.81093
[25] Kuchment, P.: Quantum graphs I. Some basic structures. Waves Random Media 14, S107–S128 (2004) · Zbl 1063.81058
[26] Kuchment, P.: Quantum graphs II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A: Math. Gen. 38, 4887–4900 (2005) · Zbl 1070.81062
[27] Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78, 201–246 (1980) · Zbl 0449.60048
[28] Levitan, B.M., Sargsyan, I.S.: Sturm-Liouville and Dirac Operators. Kluwer, Dordrecht (1990)
[29] Pankrashkin, K.: Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction. J. Math. Phys. 47, 112105 (2006) · Zbl 1112.82032
[30] Pankrashkin, K.: Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77, 139–154 (2006) · Zbl 1113.81056
[31] Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Springer, Berlin (1992) · Zbl 0752.47002
[32] Posilicano, A.: Self-adjoint extensions of restrictions. Preprint arXiv:math-ph/0703078 · Zbl 1175.47025
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