Anderson localization for the almost Mathieu equation, III. Semi-uniform localization, continuity of gaps, and measure of the spectrum

Communications in Mathematical Physics

Volumn 195, Issue 1, 1998, Pages 1-14

  Jitomirskaya, Svetlana Ya ^a   Last, Yoram ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b California Institute of Technology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032121409     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050376     Document Type: Article

References (33)

1. Aubry, G., Andre, G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc. 3, 133-140 (1980)
2. Aizenman, M.: Localization at weak disorder: Some elementary bounds. Rev. Math. Phys. 6, 1163-1182 (1994)
3. Avron, J., van Mouche, P., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys. 132, 103-118 (1990)
4. Cornfeld, I., Fomin, S., Sinai, Ya.: Ergodic Theory. New York: Springer, 1982
5. Cycon, H.L., Froese, R.G., Kirsch. W., Simon, B.: Schrödinger Operators. Berlin, Heidelberg, New York: Springer 1987
6. del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: What is localization? Phys. Rev. Lett. 75, 117-119 (1995)
7. del Rio, R., Jitomirskaya, S., Last, Y., Simon, B.: Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank-one perturbations, and localization. J. d'Analyse Math. 69, 153-200(1996)
8. del Rio, R., Jitomirskaya, S., Makarov, N., Simon, B.: Singular continuous spectrum is generic. Bull. AMS 31, 208-212 (1994)
9. del Rio, R., Makarov, N., Simon, B.: Operators with singular continuous spectrum, II. Rank one operators. Commun. Math. Phys. 165, 59-67 (1994)
10. Delyon, F., Kunz, H., Souillard, B.: One dimensional wave equations in disordered media. J. Phys. A 16, 25-42 (1983)
11. Figotin, A., Pastur, L.: An exactly solvable model of a multidimensional incommensurate structure. Commun. Math. Phys. 95, 401-425 (1984)
12. Fishman, S., Grempel, D., Prange, R.: Localization in a d-dimensional incommensurate structure. Phys. Rev. B 29, 4272-4276 (1984)
13. Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132, 5-25 (1990)
14. Gordon, A. Y: Pure point spectrum under 1-parameter perturbations and instability of Anderson localization. Commun. Math. Phys. 164, 489-505 (1994)
15. Gordon, A., Jitomirskaya, S., Last, Y., Simon B.: Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178, 169-183 (1997)
16. Jitomirskaya, S.: Anderson localization for the almost Mathieu equation; A nonperturbative proof. Commun. Math. Phys. 165, 49-58 (1994)
17. Jitomirskaya, S., Simon, B.: Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators. Commun. Math. Phys. 165, 201-205 (1994)
18. Jitomirskaya, S.: Almost everything about the almost Mathieu operator II. Proceedings of XI International Congress of Mathematical Physics, Paris 1994, Int. Press, 1995, pp. 373-382
19. Jitomirskaya, S., Last, Y: Dimensional Hausdorff properties of singular continuous spectra. Phys. Rev. Lett. 76, 1765-1769(1996)
20. Jitomirskaya, S.: Continuous spectrum and uniform localization for ergodic Schrödinger operators. J. Funct. Anal. 145, 312-322 (1997)
21. Jitomirskaya, S.: Almost Mathieu equations with weakly Liouville frequencies: Second critical constants. In preparation
22. Jitomirskaya, S., Last, Y: Power-Law subordinacy and singular spectra, II. Line operators. In preparation
23. Kunz, H., Souillard, B.: Sur le spectre des operateurs aux differences finies, aleatoires. Commun. Math. Phys. 78, 201-246 (1980)
24. Last, Y.: A relation between absolutely continuous spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Commun. Math. Phys. 151, 183-192 (1993)
25. Last, Y.: Zero measure spectrum for the almost Mathieu operator. Commun. Math. Phys. 164, 421-432 (1994)
26. Last, Y.: Almost everything about the almost Mathieu operator I. Proceedings of XI International Congress of Mathematical Physics, Paris 1994, Int. Press, 1995, pp. 366-372
27. Last Y.: Quantum dynamics and decompositions of singular continuous spectra. J. Funct. Anal. 142, 406-445 (1996)
28. Martinelli, F., Scoppola, E.: Introduction to the mathematical theory of Anderson localization. Rivista del Nuovo Cimento 10, N. 10 (1987)
29. Prange, R., Grempel, D.. Fishman, S.: A solvable model of quantum motion in an incommensurate potential. Phys. Rev. B 29, 6500-6512 (1984)
30. Simon, B.: Almost periodic Schrödinger operators, IV. The Maryland model. Ann, Phys. 159, 157-183 (1985)
31. Simon, B.: Absence of ballistic motion. Commun. Math. Phys. 134, 209-212 (1990)
32. Sinai, Ya.: Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential. J. Stat. Phys. 46, 861-909 (1987)
33. Thouless, D.J.: Bandwidth for a quasiperiodic tight binding model. Phys. Rev. 828,4272-4276(1983)