Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues

Geometric and Functional Analysis

Volumn 18, Issue 3, 2008, Pages 755-869

  Goldstein, Michael ^a   Schlag, Wilhelm ^b  

^a UNIVERSITY OF TORONTO   (Canada)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

Analytic functions; Avalanche principle; Eigenfunctions; Eigenvalues; Large deviations; Resonances

Indexed keywords

EID: 52449118491     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00039-008-0670-y     Document Type: Article

References (43)

1. [AM] M. AIZENMAN, S. MOLCHANOV, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), 245-278.
2. [An] P. ANDERSON, Absence of diffusion in certain random lattices, Phys. Rev. 109 (1958), 1492-1501.
3. [B] R. BHATIA, Perturbation Bounds for Matrix Eigenvalues, Pitman Research Notes in Mathematics Series 162, Longman, 1987.
4. [Bo1] J. BOURGAIN, Hölder regularity of integrated density of states for the almost Mathieu operator in a perturbative regime, Lett. Math. Phys. 51:2 (2000), 83-118.
5. [Bo2] J. BOURGAIN, Green's Function Estimates for Lattice Schrödinger operators and applications, Princeton University Press, 2004.
6. [BoG] J. BOURGAIN, M. GOLDSTEIN, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152:3 (2000), 835-879.
7. [BoGS] J. BOURGAIN, M. GOLDSTEIN, W. SCHLAG, Anderson localization for Schrödinger operators on ℤ with potentials given by the skew-shift, Comm. Math. Phys. 220:3 (2001), 583-621.
8. [BoS] J. BOURGAIN, W. SCHLAG, Anderson localization for Schrödinger operators on ℤ with strongly mixing potentials, Comm. Math. Phys. 215:1 (2000), 143-175.
9. [BovCKP] A. BOVIER, M. CAMPANINO, A. KLEIN, J.F. PEREZ, Smoothness of the density of states in the Anderson model at high disorder, Comm. Math. Phys. 114:4 (1988), 439-461.
10. [CK] M. CAMPANINO, A. KLEIN, A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model, Comm. Math. Phys. 104 (1986), 227-241.
11. [CaKM] R. CARMONA, A. KLEIN, F. MARTINELLI, Anderson localization for Bernoulli and other singular potentials, Comm. Math. Phys. 108:1 (1987), 41-66.
12. [CaL] R. CARMONA, J. LACROIX, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
13. [Ch] J. CHAN, Method of variations of potential of quasi-periodic Schrödinger equation, preprint 2005.
14. [CyFKS] H.L. CYCON, R.G. FROESE, W. KIRSCH, B. SIMON, Schrödinger Operators, Springer Verlag, Berlin, Heidelberg, New York, 1987.
15. [DS] E.I. DINABURG, Y.G. SINAI, The one dimensional Schrödinger equation with quasiperiodic potential, Funkt. Anal. i. Priloz. 9 (1975), 8-21.
16. [E] H. ELIASSON, Floquet solutions for the 1-dimensional quasiperiodic Schrödinger equation, Comm. Math. Phys. 146 (1992), 447-482.
17. [F] H. FEDERER, Geometric Measure Theory, Springer Die Grundlehren der mathematischen Wissenschaften 153 (1969).
18. [FiP] A. FIGOTIN, L. PASTUR, Spectra of random and almost-periodic operators, Springer Grundlehren der mathematischen Wissenschaften 297 (1992).
19. [FrS1] J. FRÖHLICH, T. SPENCER, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys. 88 (1983), 151-189.
20. [FrS2] J. FRÖHLICH, T. SPENCER, A rigorous approach to Anderson localization, Phys. Rep. 103:1-4 (1984), 9-25.
21. [FrSW] J. FRÖHLICH, T. SPENCER, P. WITTWER, Localization for a class of one dimensional quasi-periodic Schrödinger operators, Commun. Math. Phys. 132 (1990), 5-25.
22. [Fü] H. FÜRSTENBERG, Noncommuting random products, Trans. AMS 108 (1963), 377-428.
23. [FüK] H. FÜRSTENBERG, H. KESTEN, Products of random matrices, Ann. Math. Statist. 31 (1960), 457-469.
24. [GMP] I.Y. GOLDSHEID, S.A. MOLCHANOV, L.A. PASTUR, A pure point spectrum of the stochastic one-dimensional Schrödinger equation, Funkt. Anal. Appl. 11 (1977), 1-10.
25. [GoS1] M. GOLDSTEIN, W. SCHLAG, Hölder continuity of the integrated density of states for quasiperiodic Schrödinger equations and averages of shifts of subharmonic functions, Ann. of Math. (2) 154:1 (2001), 155-203.
26. [GoS2] M. GOLDSTEIN, W. SCHLAG, On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations, preprint 2005.
27. [GorJLS] A. GORDON, S. JITOMIRSKAYA, Y. LAST, B. SIMON, Duality and singular continuous spectrum in the almost Mathieu equation, Acta Math. 178 (1997), 169-183.
28. [H] P.G. HARPER, Single band motion of conduction electrons in a uniform magnetic field, Proc. Phys. Soc. London A68 (1955), 874-892.
29. [He] M. HERMAN, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le charactère local d'un theoreme d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv. 58:3 (1983), 453-502.
30. [J] S. JITOMIRSKAYA, Metal-insulator transition for the almost Mathieu operator, Annals of Math. 150:3 (1999), 1159-1175.
31. [K] A.Y. KHINCHIN, Continued Fractions, Dover, 1992.
32. [Ko] P. KOOSIS, The logarithmic integral II, Cambridge Studies in Advanced Mathematics 21, Cambridge University Press, Cambridge, 1992.
33. [L] S. LANG, Algebra, Third Edition, Addison Wesley, 1993.
34. [La1] Y. LAST, A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants, Comm. Math. Phys. 151 (1993), 183-192.
35. [La2] Y. LAST, Zero measure spectrum for the almost Mathieu operator, Comm. Math. Phys. 164 (1994), 421-432.
36. [Le] B.Y. LEVIN, Lectures on Entire Functions, Transl. of Math. Monographs 150, AMS, Providence, RI, 1996.
37. [P] J. PUIG, Cantor spectrum for the almost Mathieu operator, Comm. Math. Phys. 244:2 (2004), 297-309.
38. [ST] B. SIMON, M. TAYLOR, Harmonic analysis on SL(2,R) and smoothness of the density of states in the one-dimensional Anderson model, Comm. Math. Phys. 101:1 (1985), 1-19.
39. [Si1] Y.G. SINAI, Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential, J. Stat. Phys. 46 (1987), 861-909.
40. [Si2] Y.G. SINAI, Topics in Ergodic Theory, Princeton Mathematical Series 44. Princeton University Press, Princeton, 1994.
41. [SoS] E. SORETS, T. SPENCER, Positive Laypunov exponents for Schrödinger operators with quasi-periodic potentials, Comm.Math. Phys. 142:3 (1991), 543-566.
42. [T] D. THOULESS, Scaling for the discrete Mathieu equation, Comm. Math. Phys. 127 (1990), 187-193.
43. [W] F. WEGNER, Bounds on the density of states in disordered systems. Z. Phys. B44 (1981), 9-15.