New characterizations of the region of complete localization for random schrödinger operators

Journal of Statistical Physics

Volumn 122, Issue 1, 2006, Pages 73-94

  Germinet, Francois ^a   Klein, Abel ^b  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33644530394     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-005-8068-9     Document Type: Article

References (57)

1. Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6: 1163-1182 (1994).
2. Aizenman, M., Elgart, A., Naboko, S., Schenker, J. H., Stolz, G.: Moment analysis for localization in random schrödinger operators. Invent. Math. (2005) In press.
3. Aizenman, M., Graf, G. M.: Localization bounds for an electron gas. J. Phys. A: Math. Gen. 31: 6783-6806, (1998).
4. Aizenman, M., Molchanov, S.: Localization at large disorder and extreme energies: an elementary derivation. Commun. Math. Phys. 157: 245-278 (1993).
5. Aizenman, M., Schenker, J., Friedrich, R., Hundertmark, D.: Finite volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. 224: 219-253 (2001).
6. Bellissard, J., van Elst, A., Schulz-Baldes, H.: The non commutative geometry of the quantum Hall effect. J. Math. Phys. 35: 5373-5451 (1994).
7. Barbaroux, J. M., Combes, J. M., Hislop, P. D.: Localization near band edges for random Schrödinger operators. Helv. Phys. Acta 70: 16-43 (1997).
8. Bouclet, J. M., Germinet, F., Klein, A.: Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators. Proc. Amer. Math. Soc. 132: 2703-2712 (2004).
9. Bouclet, J. M., Germinet, F., Klein, A., Schenker, J. H.: Linear response theory for magnetic Schrdinger operators in disordered media. J. Funct. Anal. 226: 301-372 (2005).
10. Carmona, R., Klein, A., Martinelli, F.: Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108: 41-66 (1987).
11. Combes, J. M., Germinet, F.: Edge and impurity effects on quantization of hall currents. Commun. Math. Phys. 256: 159-180 (2005).
12. Combes, J. M., Germinet, F., Hislop, P.: On the quantization of Hall currents in presence of disorder, to appear in the Proceedings of the Conference Q-Math9 (Giens, 2004).
13. Combes, J. M., Hislop, P. D.: Localization for some continuous, random Hamiltonian in d-dimension. J. Funct. Anal. 124: 149-180 (1994).
14. Combes, J. M., Hislop, P. D.: Landau Hamiltonians with random potentials: localization and the density of states. Commun. Math. Phys. 177: 603-629 (1996).
15. Combes, J. M., Hislop, P. D., Tip, A.: Band edge localization and the density of states for acoustic and electromagnetic waves in random media. Ann. Inst. H. Poincare Phys. Theor. 70: 381-428 (1999).
16. Damanik, D., Sims, R., Stolz, G.: Localization for one dimensional, continuum, BernoulliAnderson models. Duke Math. J. 114: 59-100 (2002).
17. Damanik, D., Stollmann, P.: Multi-scale analysis implies strong dynamical localization. Geom. Funct. Anal. 11: 11-29 (2001).
18. Del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: What is Localization? Phys. Rev. Lett. 75: 117-119 (1995).
19. Del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum IV: Hausdorff dimensions, rank one pertubations and localization. J. d'Analyse Math. 69: 153-200 (1996).
20. Dobrushin, R., Shlosman, S.: Completely analytical Gibbs fields. Prog in Phys. 10: 347-370 (1985).
21. Dobrushin, R., Shlosman, S.: Completely analytical interactions. J. Stat. Phys. 46: 983-1014 (1987).
22. von Dreifus, H.: On the effects of randomness in ferromagnetic models and Schrödinger operators. Ph.D. thesis, New York University (1987)
23. von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124: 285-299 (1989).
24. Elgart, A., Graf, G. M., Schenker, J. H.: Equality of the bulk and edge hall conductances in a mobility gap. Commun Math. Phys (2005) In press.
25. Figotin, A., Klein, A.: Localization phenomenon in gaps of the spectrum of random lattice operators. J. Stat. Phys. 75: 997-1021 (1994).
26. Figotin, A., Klein, A.: Localization of electromagnetic and acoustic waves in random media. Lattice model. J. Stat. Phys. 76: 985-1003 (1994).
27. Figotin, A., Klein, A.: Localization of classical waves I: Acoustic waves. Commun. Math. Phys. 180: 439-482 (1996).
28. Figotin, A., Klein, A.: Localization of classical waves II: Electromagnetic waves. Commun. Math. Phys. 184: 411-441 (1997).
29. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101: 21-46 (1985).
30. Fröhlich, J., Spencer, T.: Absence of diffusion with Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88: 151-184 (1983).
31. Germinet, F.: Dynamical localization II with an application to the almost Mathieu operator. J. Stat Phys. 95: 273-286 (1999).
32. Germinet, F., De Bièvre, S.: Dynamical localization for discrete and continuous random Schrödinger operators. Commun. Math. Phys. 194: 323-341 (1998).
33. Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222: 415-448 (2001).
34. Germinet, F., Klein, A.: Operator kernel estimates for functions of generalized Schrödinger operators. Proc. Amer. Math. Soc. 131: 911-920 (2003).
35. Germinet, F, Klein, A.: Explicit finite volume criteria for localization in continuous random media and applications. Geom. Funct. Anal. 13: 1201-1238 (2003).
36. Germinet, F., Klein, A.: High disorder localization for random Schrödinger operators through explicit finite volume criteria. Markov Process. Related Fields. 9: 633-650 (2003).
37. Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport transition. Duke Math. J. 124: 309-351 (2004).
38. Germinet, F., Klein, A.: Localization zoology for Schrödinger operators. In preparation.
39. Germinet, F, Klein, A., Schenker, J.: Dynamical delocalization in random Landau Hamiltonians. To appear in Annals of Math.
40. Kirsch, W., Stollman, P., Stolz, G.: Localization for random perturbations of periodic Schrödinger operators. Random Oper. Stochastic Equations 6: 241-268 (1998).
41. Klein, A.: Multiscale analysis and localization of random operators. In Random Schrodinger operators: methods, results, and perspectives. Panorama & Synthèse, Société Mathématique de France. To appear.
42. Klein, A., Koines, A., Seifert, M.: Generalized eigenfunctions for waves in inhomogeneous media. J. Funct. Anal. 190: 255-291 (2002).
43. Klein, A., Koines, A.: A general framework for localization of classical waves: II. Random media. Math. Phys. Anal. Geom. 7: 151-185 (2004).
44. Klein, A., Lacroix, J., Speis, A.: Localization for the Anderson model on a strip with singular potentials. J. Funct. Anal. 94: 135-155 (1990).
45. Klein, A., Molchanov, S.: Simplicity of eigenvalues in the Anderson model. J. Stat. Phys. (2005) In press.
46. Klopp, F.: Localization for continuous random Schrödinger operators. Commun. Math. Phys. 167: 553-569 (1995).
47. Klopp, F.: Weak disorder localization and Lifshitz tails. Commun. Math. Phys. 232: 125-155 (2002).
48. Klopp, F.: Weak disorder localization and Lifshitz tails: continuous Hamiltonians. Ann. I. H. P. 3: 711-737 (2002).
49. Minami, N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Comm. Math. Phys. 177: 709-725 (1996).
50. Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Heidelberg: Springer-Verlag, 1992
51. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged edition. Academic Press, 1980.
52. Simon, B.: Cyclic vectors in the Anderson model. Special issue dedicated to Elliott H. Lieb. Rev. Math. Phys. 6: 1183-1185 (1994).
53. Spencer, T.: Localization for random and quasiperiodic potentials. J. Stat. Phys. 51: 1009-1019 (1988).
54. Stollmann, P.: Caught by disorder. Bound States in Random Media. Birkaüser 2001.
55. Tcheremchantsev, S.: How to prove dynamical localization, Commun. Math. Phys. 221: 27-56 (2001).
56. Wang, W.-M.: Microlocalization, percolation, and Anderson localization for the magnetic Schrödinger operator with a random potential. J. Funct. Anal. 146: 1-26 (1997).
57. Wang, W.-M.: Localization and universality of Poisson statistics for the multidimensional Anderson model at weak disorder. Invent. Math. 146: 365-398 (2001).