The integrated density of states for some random operators with nonsign definite potentials

Journal of Functional Analysis

Volumn 195, Issue 1, 2002, Pages 12-47

  Hislop, Peter D ^a   Klopp, Frédéric ^b  

^a UNIVERSITY OF KENTUCKY   (United States)

^b University Sorbonne Paris Nord   (France)

Author keywords

Localization; Monotonic variation; Schr dinger operators; Wegner estimate

Indexed keywords

EID: 0037145490     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.2002.3947     Document Type: Article

References (45)

1. M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209-273.
2. J.-M. Barbaroux, J. M. Combes, and P. D. Hislop, Localization near band edges for random Schrödinger operators, Helv. Phys. Acta 70 (1997), 16-43.
3. M. S. Birman and M. Z. Solomjak, Remarks on the spectral shift function, J. Soviet Math. 3 (1975), 408-419.
4. M. S. Birman and M. Z. Solomjak, "Spectral Theory of Self-Adjoint Operators in Hilbert Space," D. Reidel Publishing Co., Dordrecht, 1987.
5. M. S. Birman and D. R. Yafaev, The spectral shift function: The work of M. G. Krein and its further development, St. Petersburg Math. J. 4 (1992), 833-870.
6. Ph. Briet and H. D. Cornean, Locating the spectrum for magnetic Schrödinger and Dirac operators, Commun. PDE 27 (2002), 1079-1101.
7. Deleted in proof.
8. J. M. Combes and P. D. Hislop, Localization for some continuous random Hamiltonians in d-dimensions, J. Funct. Anal. 124 (1994), 149-180.
9. J. M. Combes, P. D. Hislop, and E. Mourre, Spectral averaging, perturbation of singular spectra, and localization, Trans. Amer. Math. Soc. 348 (1996), 4883-4894.
10. J. M. Combes, P. D. Hislop, and E. Mourre, Correlated Wegner inequalities for random Schrödinger operators, in "Advances in Differential Equations and Mathematical Physics: Proceedings of the International Conference on Mathematical Physics" (E. Carlen, E. M. Harrell, and M. Loss, Eds.), Contemporary Mathematics, Vol. 217, pp. 191-204, American Math. Soc., Providence, RI, 1998.
11. p-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators, Comm. Math. Phys. 218 (2001), 113-130.
12. J. M. Combes, P. D. Hislop, and A. Tip, Band edge localization and the density of states for acoustic and electromagnetic waves in random media, Ann. Inst. H. Poincaré 70 (1999), 381-428.
13. J. M. Combes and L. Thomas, Asymptotic behavior of eigenfunctions for multiparticle Schrödinger operators, Comm. Math. Phys. 34 (1973), 251-270.
14. S. Doi, A. Iwatsuka, and T. Mine, The uniqueness of the integrated density of states for Schrödinger operators with magnetic fields, Math. Z. 237 (2001), 335-371.
15. A. Figotin and A. Klein, Localization of classical waves I: Acoustic waves, Comm. Math. Phys. 180 (1996), 439-482.16
16. A. Figotin and A. Klein, Localization of classical waves II: Electromagnetic waves, Comm. Math. Phys. 184 (1997), 411-441.
17. F. Ghribi, Thesis, Université Paris XIII, in preparation.
18. R. Hempel and I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Comm. Math. Phys. 169 (1995), 237-259.
19. p-bound on the Krein spectral shift function, J. d'Analyse Math., to appear.
20. T. Hupfer, H. Leschke, P. Müller, and S. Warzel, The existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev. Math. Phys. 13 (2001), 1547-1581.
21. W. Kirsch, Random Schrödinger operators: a course, in "Schrödinger Operators" (H. Holden and A. Jensen, Eds.), Lecture Notes in Physics, Vol. 345, Sonderborg, Denmark 1988, Springer, Berlin, 1989.
22. W. Kirsch, Wegner estimates and localization for alloy-type potentials, Math. Z. 221 (1996), 507-512.
23. W. Kirsch, P. Stollmann, and G. Stolz, Localization for random perturbations of periodic Schrödinger operators, Random Operators Stochastic Equations 6 (1998), 241-268.
24. W. Kirsch, P. Stollmann, and G. Stolz, Anderson localization for random Schrödinger operators with long range interactions, Comm. Math. Phys. 195 (1998), 495-507.
25. F. Klopp, Localization for some continuous random Schrödinger operators, Comm. Math. Phys. 167 (1995), 553-569.
26. F. Klopp, Internal Lifshitz tails for random perturbations of periodic Schrödinger operators, Duke Math. J. 98 (1999), 335-396.
27. F. Klopp and T. Wolff, Lifshitz tails for 2-dimensional random Schrödinger operators, J. d'Analyse Math., to appear.
28. V. Kostrykin and R. Schrader, Scattering theory approach to random Schrödinger operators in one-dimension, Rev. Math. Phys. 11 (1999), 187-242.
29. V. Kostrykin and R. Schrader, The density of states and the spectral shift density of random Schrödinger operators, Rev. Math. Phys. 12 (2000), 807-848.
30. V. Kostrykin and R. Schrader, Global bounds for the Lyapunov exponent and the integrated density of states of random Schrödinger operators in one dimension, J. Phys. A 33 (2000), 8231-8240.
31. S. Kotani and B. Simon, Localization in general one dimensional systems. II, Comm. Math. Phys. 112 (1987), 103-120.
32. S. Nakamura, Band spectrum for Schrödinger operators with strong periodic magnetic fields, in "Operator Theory: Advances and Applicationg" (M. Demuth and B.-W. Schulze, Eds.), Vol. 78, Birkhäuser Verlag, Basel, 1995.
33. S. Nakamura, Lifshitz tail for Schrödinger operator with random magnetic field, Comm. Math. Phys. 214 (2000), 565-572.
34. S. Nakamura, A remark on the Dirichlet-Neumann decoupling and the integrated density of states, J. Funct. Anal. 179 (2001), 136-152.
35. B. Simon, "Trace Ideals and their Applications," London Mathematical Society Lecture Series, Vol. 35, Cambridge Univ. Press, Cambridge, UK, 1979.
36. B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. N. S. 7 (1982), 447-526; Errata, Bull. Amer. Math. Soc. N. S. 11 (1984), 426.
37. B. Simon, Spectral averaging and the Krein spectral shift, Proc. Amer. Math. Soc. 126 (1998), 1409-1413.38
38. Deleted in proof.
39. P. Stollmann, Wegner estimates and localization for continuum Anderson models with some singular distributions, Arch. Math. 75 (2000), 307-311.
40. P. Stollmann, Caught by disorder: Bound states in random media, Progr. Math. Phys. 20 (2001).
41. G. Stolz, Non-monotonic random Schrödinger operators: The Anderson model, J. Math. Anal. Appl. 248 (2000), 173-183.
42. I. Veselić, Wegner estimate for some Anderson-type Schrödinger operators, preprint 2000.
43. H. von Dreifus and A. Klein, A new proof of localization for the Anderson tigbt-binding model, Comm. Math. Phys. 140 (1991), 133-147.
44. F. Wegner, The density of states for disordered systems, Z. Phys. B 44 (1981), 9-15.
45. D. R. Yafaev, "Mathematical Scattering Theory: General Theory," Translations of Mathematical Monographs, Vol. 105, Amer. Math. Soc., Providence, RI. 1992.