Spectral localization by Gaussian random potentials in multi-dimensional continuous space

Journal of Statistical Physics

Volumn 101, Issue 5-6, 2000, Pages 935-985

  Fischer, Werner ^a,b   Leschke, Hajo ^a   Müller, Peter ^c  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^b INFINEON TECHNOLOGIES AG   (Germany)

^c UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Anderson localization; Random Schr dinger operators

Indexed keywords

EID: 0034549703     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1026425621261     Document Type: Article

References (87)

1. R. J. Adler, The Geometry of Random Fields (Wiley, Chichester, 1981).
2. M. Aizenman, Localization at weak disorder: Some elementary bounds. Rev. Math. Phys. 6:1163-1182 (1994).
3. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109:1492-1505 (1958).
4. M. Aizenman and G. M. Graf. Localization bounds for an electron gas, J. Phys. A 31:6783-6806 (1998).
5. M. Aizenman and S. Molchanov, Localizalion at large disorder and at extreme energies: An elementary derivation, Commun. Math. Phys. 157:245-278 (1993).
6. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, ninth printing (Dover Publications. New York, 1972).
7. M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Commun. Pure Appl. Math. 35:209-273 (1982).
8. M. Aizenman, J. H. Schenker, R. M. Friedrich, and D. Hundertmark, Finite-volume criteria for Anderson localization. e-print math-ph 9910022 (1999), to appear in Commun. Math. Phys.
9. Yu. K. Belyaev, Analytic random processses, Theory Prob. Applications 4:402-409 (1959) [Russian original: Teor. Ver. Primen. 4:437-444 (1959)].
10. Yu. K. Belyaev, Analytic random processses, Theory Prob. Applications 4:402-409 (1959) [Russian original: Teor. Ver. Primen. 4:437-444 (1959)].
11. S. M. Berman, The maximum of a Gaussian process with nonconstant variance. Ann. Inst. Henri Poincaré. Probab. Stat. 21:383-391 (1985).
12. J.-M. Barbaroux, J. M. Combes, and P. D. Hislop, Landau Hamiltonians with unbounded random potentials. Lett. Math. Phys. 40:355-369 (1997).
13. J. M. Barbaroux, J. M. Combes, and P. D. Hislop, Localization near band edges for random Schrödinger operators, Helv. Phys. Acta 70:16-43 (1997).
14. J. M. Barbaroux, J. M. Combes, and R. Monteho, Remarks on the relation between quantum dynamics and fractal spectra, J. Math. Anal. Appl. 213:698-722 (1997).
15. V. L. Bonch-Bruevich, R. Enderlein, B. Esser, R. Keiper, A. G. Mironov, and I. P. Zvyagin, Elektronentheorie ungeordneter Halbleiter (VEB Deutscher Verlag der Wissenschaften, Berlin. 1984) [Russian original: (Nauka, Moscow, 1981)].
16. J. M. Barbaroux, W. Fischer, and P. Müller, Dynamical properties of random Schrödinger operators, e-print math-ph 9907002 (1999).
17. A. Boutet de Monvel and V. Grinshpun, Exponential localization for multi-dimensional Schrödinger operator with random point potential, Rev. Math. Phys. 9:425-451 (1997).
18. K. Broderix, D. Hundertmark, and H. Leschke, Self-averaging, decomposition and asymptotic properties of the density of states for random Schrödinger operators with constant magnetic field, in Path Integrals from meV to MeV: Tutzing '92, H. Grabert, A. Inomata, L. S. Schulman, and U. Weiss, eds. (World Scientific. Singapore, 1993), pp. 98-107.
19. K. Broderix, D. Hundertmark, and H. Leschke, Continuity properties of Schrödinger semigroups with magnetic fields, Rev. Math. Phys. 12:181-225 (2000).
20. D. Buschmann and G. Stolz, Two-parameter spectral averaging and localization for non-monotonous random Schrödinger operators (1998), to appear in Trans. Amer. Math. Soc.
21. H. L. Cycon, R. G. Frocse, W. Kirsch, and B. Simon, Schrödinger Operators (Springer, Berlin, 1987).
22. J.-M. Combes and P. D. Hislop, Localization for some continuous, random Hamiltonians in d-dimensions, J. Funet. Anal. 124:149-180 (1994).
23. J. M. Combes and P. D. Hislop, Landau Hamiltonians with random potentials: Localization and the density of states, Commun. Math. Phys. 177:603-629 (1996).
24. J. M. Combes, P. D. Hislop, and E. Mourre, Correlated Wegner inequalities for random Schrödinger operators. Contemporary Mathematics 217:191-203 (1998).
25. R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators (Birkhäuser, Boston, 1990).
26. J. M. Combes and L. Thomas, Asymptotic behaviour of eigenfunctions for multi-particle Schrödinger operators, Commun. Math. Phys. 34:251-270 (1973).
27. P. Doukhan, Mixing, Properties and Examples (Springer, New York, 1994).
28. H. von Dreifus and A. Klein, A new proof of localization in the Anderson tight binding model, Commun. Math. Phys. 124:285-299 (1989).
29. H. von Dreifus and A. Klein, Localization for random Schrödinger operators with correlated potentials, Commun. Math. Phys. 140:133-147 (1991).
30. F. Delyon, Y. Lévy, and B. Souillard, Anderson localization for multi-dimensional systems at large disorder or large energy, Commun. Math. Phys. 100: 463-470 (1985).
31. T. C. Dorlas, N. Macris, and J. V. Pulé, Localisation in a single-band approximation to random Schrödinger operators in a magnetic field, Helv. Phys. Acta 68: 329-364 (1995).
32. T. C. Dorlas, N. Macris, and J. V. Pulé, Localization in single Landau bands, J. Math. Phys. 37:1574-1595 (1996).
33. T. C. Dorlas, N. Macris, and J. V. Pulé. Characterization of the spectrum of the Landau Hamiltonian with delta impurities, Commun. Math. Phys. 204:367-396 (1999).
34. D. Damanik and P. Stollmann, Multi-scale analysis implies strong dynamical localization, e-print math-ph/9912002 (1999), to appear in Geom. Funet. Anal.
35. K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997).
36. X. Fernique, Regularité des trajectoires des fonctions aléatoires Gaussiennes, in Ecole d'Eté de Probabilités de Saint-Flour IV-1974, Lecture Notes in Mathematics, Vol. 480, P.-L. Hennequin, ed. (Springer, Berlin, 1975), pp. 1-96.
37. W. Fischer, T. Hupfer, H. Leschke, and P. Müller, Rigorous results on Schrödinger operators with certain Gaussian random potentials in multi-dimensional continuous space, in Differential Equations, Asymptotic Analysis, and Mathematical Physics, M. Demuth and B.-W. Schulze, eds. (Akademie, Berlin, 1997), pp. 105-112.
38. W. Fischer, T. Hupfer, H. Leschke, and P. Müller, Existence of the density of states for multi-dimensional continuum Schrödinger operators with Gaussian random potentials, Commun. Math. Phys. 190:133-141 (1997).
39. A. Figotin and A. Klein, Localization of classical waves I: Acoustic waves, Commun. Math. Phys. 180:439-482 (1996).
40. W. Fischer, H. Leschke, and P. Müller, Towards localisation by Gaussian random potentials in multi-dimensional continuous space, Lett. Math. Phys. 38: 343-348 (1996).
41. J. Fröhlich, F. Martinelli, E. Scoppola, and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Commun. Math. Phys. 101: 21-46 (1985).
42. J. Fröhlich and T. Spencer. Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. 88:151-184 (1983).
43. G. M. Graf, Anderson localization and the space-time characteristic of continuum states, J. Stat. Phys. 75:337-346 (1994).
44. A. G. Grin, On a condition for regularity of stationary random processes, Theory Prob. Applications 27:850-855 (1982) [Russian original: Teor. Ver. Primen. 27:789-795 (1982)].
45. A. G. Grin, On a condition for regularity of stationary random processes, Theory Prob. Applications 27:850-855 (1982) [Russian original: Teor. Ver. Primen. 27:789-795 (1982)].
46. F. Germinet and S. De Bièvre, Dynamical localization for discrete and continuous random Schrödinger operators, Commun. Math. Phys. 194:323-341 (1998).
47. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View. 2nd ed. (Springer, New York, 1987).
48. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. corrected and enlarged edition (Academic, San Diego, 1980).
49. J. S. Howland, Perturbation theory of dense point spectra, J. Funct. Anal. 74: 52-80 (1987).
50. D. Hundertmark, On the time-dependent approach to Anderson localization. Math. Nachr. 214:25-38 (2000).
51. T. Hupfer, H. Lesehke, P. Müller, and S. Warzel, The integrated density of states and its absolute continuity for magnetic Schrödinger operators with unbounded random potentials, e-print math-ph 0010013 (2000).
52. A. M. Hinz, and G. Stolz. Polynomial boundedness of eigensolutions and the spectrum of Schrödinger operators, Math. Ann. 294:195-211 (1992).
53. I. A. Ibragimov and Y. A. Rozanov, Gaussian Random Processes (Springer. New York, 1978) [Russian original: (Nauka, Moscow, 1970)].
54. O. Kallenberg, Foundations of Modern Probability (Springer, New York, 1997).
55. W. Kirsch, Random Schrödinger operators: A course, in Schrödinger Operators, Lecture Notes in Physics. Vol. 345, H. Holden and A. Jensen. eds. (Springer, Berlin, 1989), pp. 264-370.
56. W. Kirsch, Wegner estimates and Anderson localization for alloy-type potentials, Math. Z. 221:507-512 (1996).
57. d): Un modèle semi-classique. Ann. Inst. Fourier 45:265-316 (1995).
58. F. Klopp, Localization for some continuous random Schrödinger operators, Commun. Math. Phys. 167:553-569 (1995).
59. A. Kiselev and Y. Last, Solutions, spectrum. and dynamics for Schrödinger operators on infinite domains, Duke Math. Journal 102:125-150 (2000).
60. W. Kirsch and F. Martinelli, On the ergodic properties of the spectrum of general random operators. J. Reine Angew. Math. 334:141-156 (1982).
61. W. Kirsch, P. Stollmann, and G. Stolz, Localization for random perturbations of periodic Schrödinger operators, Random Operators and Stochastic Equations 6:241-268 (1998).
62. W. Kirsch, P. Stollmann, and G. Stolz, Anderson localization for random Schrödinger operators with long range interactions, Commun. Math. Phys. 195:495-507 (1998).
63. Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142:406-445 (1996).
64. M. A. Lifshits, Ganssian Random Functions (Kluwer, Dordrecht, 1995).
65. I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988) [Russian original: (Nauka, Moscow, 1982)].
66. H. Matsumoto, On the integrated density of states for the Schrödinger operators with certain random electromagnetic potentials, J. Math. Soc. Japan 45:197-214 (1993).
67. v), Commun. Math. Phys. 93:197-217 (1984).
68. F. Martinelli and E. Scoppola, Remark on the absence of absolutely continuous spectrum for d-dimensional Schrödinger operators with random potential for large disorder or low energy, Commun. Math. Phys. 97:465-471 (1985). Erratum: ibid. 98:579 (1985).
69. F. Martinelli and E. Scoppola, Introduction to the mathematical theory of Anderson localization, Rivista del Nuovo Cimento 10(10):1-90 (1987).
70. L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators (Springer, Berlin, 1992).
71. J. V. Pulé and M. Scrowston, Infinite degeneracy for a Landau Hamiltonian with Poisson impurities, J. Math. Phys. 38:6304-6314 (1997).
72. J. V. Pulé and M. Scrowston, The spectrum of a magnetic Schrödinger operator with randomly located delta impurities, J. Math. Phys. 41:2805-2825 (2000).
73. R. D. Richtmyer, Principles of Advanced Mathematical Physics, Vol. I (Springer, New York, 1978).
74. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum IV: Hausdorff dimensions, rank one perturbations and localization. J. d'Analyse Math. 69:153-200 (1996).
75. M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. and enl, ed. (Academic, San Diego, 1980).
76. M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic, New York, 1975).
77. B. Simon, Absence of ballistic motion, Commun. Math. Phys. 134:209-212 (1990).
78. T. C. Spencer, The Schrödinger equation with a random potential: A mathematical review, in Critical Phenomena, Random Systems, Gauge Theories, Part II, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986), pp. 895-943.
79. T. Spencer, Localization for random and quasiperiodic potentials, J. Stat. Phys. 51:1009-1019 (1988).
80. P. Stollmann, Caught by Disorder: Lectures on Bound States in Random Media (Birkhäuser), to appear.
81. B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer, Berlin, 1984) [Russian original: (Nauka, Moscow, 1979)].
82. B. Simon and T. Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Commun. Pure Appl. Math. 39:75-90 (1986).
83. N. Ueki, On spectra of random Schrödinger operators with magnetic fields, Osaka J. Math. 31:177-187 (1994).
84. I. Veselić, Localisation for random perturbations of periodic Schrödinger operators with regular Floquet eigenvalues, e-print mp_arc 98-569 (1998).
85. Wei-Min Wang, Microlocalization, percolation, and Anderson localization for the magnetic Schrödinger operator with a random potential, J. Funct. Anal. 146:1-26 (1997).
86. V. Yurinsky, Sums and Gaussian Vectors, Lecture Notes in Mathematics. Vol. 1617 (Springer, Berlin, 1995).
87. H. Zenk, Anderson localization for a multidimensional model including long range potentials and displacements, e-print mp_arc 99-450 (1999).