On the nature of the spectrum of the quasi-periodic Schrödinger operator

Nonlinear Analysis: Real World Applications

Volumn 3, Issue 1, 2002, Pages 37-59

  Fabbri, Roberta ^a   Johnson, Russell ^a   Pavani, Raffaella ^b  

^a UNIVERSITY OF FLORENCE   (Italy)

^b POLITECNICO DI MILANO   (Italy)

Author keywords

Cantor spectrum; Quasi periodic Schrodinger operator; Rotation number

Indexed keywords

EID: 0041937763     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/S1468-1218(01)00012-8     Document Type: Article

References (33)

1. S. Aubrey, G. Andre, Analyticity breaking and Anderson localization in noncommensurate lattices, Ann. Israel Phys. Soc. 3 (1980) 133-140.
2. M. Azbel, Quantum particle in one-dimensional potential with incommensurate periods, Phys. Rev. Lett. 43 (1979) 1954-1957.
3. G. Choquet, Lectures on Analysis, Vol. I, II, III, Benjamin, New York, 1969.
4. V. Chulaevsky, Ya. Sinai, Anderson localization for the I-D discrete Schrödinger operators with two-frequency potential, Comm. Math. Phys. 125 (1989) 91-112.
5. A. Coppel, in: Dichotomies in Stability Theory, Lectures Notes in Mathematics, Vol. 629, Springer, New York, Heidelberg, Berlin, 1978.
6. C. De Concini, R. Johnson, The algebraic-geometric AKNS potentials, Eng. Theory Dynamic Systems 7 (1987) 1-24.
7. B. Dubrovin, S. Novikov, V. Matveev, Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Russ. Math. Surv. 31 (1976) 59-146.
8. L.H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation, Comm. Math. Phys. 146 (1992) 447-482.
9. L.H. Eliasson, Discrete one-dimensional quasi-periodic Schrödinger operator with pure point spectrum, Acta Math. 179 (1997) 153-196.
10. R. Fabbri, Genericitá dell'Iperbolicitá nei Sistemi Differenziali Lineari di Dimensione Due, Ph.D. Thesis, Università di Firenze, 1997.
11. R. Fabbri, R. Johnson, On the Lyapunov exponent of certain SL(2,R)-valued cocycles, Equations and Dynamical Systems 7 (1999) 349-370.
12. R. Fabbri, R. Johnson, Genericity of exponential dichotomy for two-dimensional quasi-periodic linear differential systems, Ann. Mat. Pura Appl. 178 (2000) 175-193.
13. H. Federer, Geometric Measure Theory, Springer, New York, Heidelberg, Berlin, 1967.
14. J. Fröhlich, T. Spencer, P. Wittwer, Localization for a class of one dimensional quasi-periodic Schrödinger operators, Comm. Math. Phys. 132 (1990) 5-25.
15. R. Giachetti, R. Johnson, Spectral theory of two-dimensional almost periodic differential operators and its relation to classes of nonlinear evolution equations, Il Nuovo Cimento 82 (1984) 125-168.
16. N.J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, PA, 1996.
17. D. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14 (1976) 2239-2249.
18. R. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986) 54-78.
19. R. Johnson, Cantor spectrum for the quasi-periodic Schrödinger equation, J. Differential Equations 91 (1991) 88-110.
20. R. Johnson, J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982) 403-438.
21. R. Johnson, K.J. Palmer, G.R. Sell, Ergodic properties of linear dynamical systems, Siam J. Math. Anal. 18 (1987) 1-33.
22. M. Kohmoto, M. Nightingale, M. den Nijs, D. Thouless, Quantum Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49 (1982) 405.
23. S. Kotani, Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, Proceedings of the Taniguchi Symposium S.S., Katata, 1982, pp. 225-247.
24. W. Magnus, S. Winkler, Hill's Equation, Interscience Publishers, New York, London, Sydney, 1966.
25. J. Moser, An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helvetici 56 (1981) 198-224.
26. M. Nerurkar, Positive exponents for a dense class of continuous SL(2,R)-valued cocycles which arise as solutions to strongly accessible linear differential systems, Contemp. Math. 215 (1998) 265-278.
27. R. Pavani, A numerical approximation of the rotation number, Appl. Math. Comput. 73 (1995) 103-110.
28. R. Pavani, The numerical approximation of the rotation number of planar maps, Comput. Math. Appl. 33 (1997) 103-110.
29. R. Pavani, R. Talamo, Conjugating the Poincare' map to a rotation, Ann. Mat. Pura Appl. 166 (1994) 381-394.
30. R.J. Sacker, G.R. Sell, A spectral theory for linear differential systems, J. Differential Equations 27 (1978) 320-358.
31. G. Segal, G. Wilson, Loop groups and equations of K-dV type, Publ. IHES 61 (1985) 5-65.
32. Ya. Sinai, Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, J. Statist. Phys. 46 (1987) 861-909.
33. G. Wannier, F. Claro, Magnetic subband structure of electrons in hexagonal lattices, Phys. Rev. B 19 (1979) 6068.