On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator

R. Fabbri

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 1, page 149-161
  • ISSN: 0392-4041

Abstract

top
In this paper we study the Lyapunov exponent β ⁢ E for the one-dimensional Schrödinger operator with a quasi-periodic potential. Let Γ ⊂ R k be the set of frequency vectors whose components are rationally independent. Let Γ ⊂ R k , and consider the complement in Γ ⁢ C r ⁢ T k of the set D where exponential dichotomy holds. We show that β = 0 is generic in this complement. The methods and techniques used are based on the concepts of rotation number and exponential dichotomy.

How to cite

top

Fabbri, R.. "On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator." Bollettino dell'Unione Matematica Italiana 5-B.1 (2002): 149-161. <http://eudml.org/doc/195559>.

@article{Fabbri2002,
abstract = {In this paper we study the Lyapunov exponent $\beta(E)$ for the one-dimensional Schrödinger operator with a quasi-periodic potential. Let $\Gamma\subset \mathbb\{R\}^\{k\}$ be the set of frequency vectors whose components are rationally independent. Let $\Gamma\subset \mathbb\{R\}^\{k\}$, and consider the complement in $\Gamma \times C^\{r\} (\mathbb\{T\}^\{k\} )$ of the set $\mathcal\{D\}$ where exponential dichotomy holds. We show that $\beta=0$ is generic in this complement. The methods and techniques used are based on the concepts of rotation number and exponential dichotomy.},
author = {Fabbri, R.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {149-161},
publisher = {Unione Matematica Italiana},
title = {On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator},
url = {http://eudml.org/doc/195559},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Fabbri, R.
TI - On the Lyapunov exponent and exponential dichotomy for the quasi-periodic Schrödinger operator
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/2//
PB - Unione Matematica Italiana
VL - 5-B
IS - 1
SP - 149
EP - 161
AB - In this paper we study the Lyapunov exponent $\beta(E)$ for the one-dimensional Schrödinger operator with a quasi-periodic potential. Let $\Gamma\subset \mathbb{R}^{k}$ be the set of frequency vectors whose components are rationally independent. Let $\Gamma\subset \mathbb{R}^{k}$, and consider the complement in $\Gamma \times C^{r} (\mathbb{T}^{k} )$ of the set $\mathcal{D}$ where exponential dichotomy holds. We show that $\beta=0$ is generic in this complement. The methods and techniques used are based on the concepts of rotation number and exponential dichotomy.
LA - eng
UR - http://eudml.org/doc/195559
ER -

References

top
  1. CHOQUET, G., Lectures on Analysis, vol I, II, III. BenjaminN.Y., 1969. Zbl0181.39601
  2. CHULAEVSKY, V.- SINAI, YA., Anderson localization for the 1-D discrete Schrödinger operators with two-frequency potential, Comm. Math. Phys, 125 (1989), 91-112. Zbl0743.60058MR1017741
  3. COPPEL, A., Dichotomies in Stability Theory, Lectures Notes in Mathematics, 629, Springer-Verlag, New York/Heidelberg/Berlin (1978). Zbl0376.34001MR481196
  4. DE CONCINI, C.- JOHNSON, R., The algebraic-geometric AKNS potentials, Erg. Th. Dyn. Sys.7 (1992), 1-24. Zbl0636.35077MR886368
  5. ELIASSON, L.H., Floquet solutions for the 1-dimensional quasi-periodic Schrodinger equation, Commun. Math. Phys., 146 (1992), 447-482. Zbl0753.34055MR1167299
  6. ELIASSON, L.H., Discrete one-dimensional quasi-periodic Schrodinger operator with pure point spectrum, Acta Math., 179 (1997), 153-196. Zbl0908.34072MR1607554
  7. FABBRI, R., Genericità dell'Iperbolicita nei Sistemi Differenziali Lineari di Dimensione Due, Ph.D. Thesis, Università di Firenze, 1997. 
  8. FABBRI, R.- JOHNSON, R., On the Lyapunov exponent of certain S ⁢ L ⁢ 2 , R -valued cocycles, Diff. Eqns. and Dynam. Sys., 7, 3 (1999), 349-370. Zbl0989.34041MR1861078
  9. FABBRI, R.- JOHNSON, R., Genericity of exponential dichotomy for two-dimensional quasi-periodic linear differential systems, Ann. Mat. Pura ed Appl., 178 (2000), 175-193. Zbl1037.34043MR1849385
  10. FEDERER, H., Geometric Measure Theory, Springer-Verlag, New York / Heidelberg / Berlin, 1967. Zbl0176.00801
  11. FABBRI, R.- JOHNSON, R.- PAVANI, R., On the Nature of the Spectrum of the Quasi-Periodic Schrodinger Operator, Nonlinear Analysis RWA, 3 (2001), 37-59. Zbl1036.34097MR1941947
  12. FRÖHLICH, J.- SPENCER, T.- WITTWER, P., Localization for a class of one dimensional quasi-periodic Schrodinger operators, Commun. Math. Phys., 132 (1990), 5-25. Zbl0722.34070MR1069198
  13. GIACHETTI, R.- JOHNSON, R., 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. MR770735
  14. HERMAN, M., Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractèr local d¡¯un théorèm d'Arnold et de Moser sur le tore en dimension 2, Commun. Math. Helv., 58 (1983), 453-502. Zbl0554.58034MR727713
  15. JOHNSON, R., Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, Journal of Differential Equations, 61 (1986), 54-78. Zbl0608.34056MR818861
  16. JOHNSON, R., Cantor spectrum for the quasi-periodic Schrodinger equation, Journal of Differential Equations, 91 (1991), 88-110. Zbl0734.34074MR1106119
  17. JOHNSON, R.- MOSER, J., The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. Zbl0497.35026MR667409
  18. JOHNSON, R.- PALMER, K. J.- SELL, G. R., Ergodic properties of linear dynamical systems, Siam J. Math. Anal., 18 (1987), 1-33. Zbl0641.58034MR871817
  19. KOTANI, S., Lyapunov indices determine absolutely continuous spectra of stationary random one-dimensional Scrodinger operators, Proc. Taniguchi Symp. S.S., Katata (1982), 225-247. Zbl0549.60058MR780760
  20. MAGNUS, W.- WINKLER, S., Hill's Equation, New York-London-Sydney, Interscience Publishers, 1966. Zbl0158.09604MR197830
  21. MAÑÉ, R., OSELEDEC'S THEOREM FROM THE GENERIC VIEWPOINT, PROC. INT. CONGR. MATH. 1983, WARSAW, 1296-1276. Zbl0584.58007MR804776
  22. MILLIONŠČIKOV, V., Proof of the existence...almost periodic coefficients, Differential Equations, 4 (1968), 203-205. Zbl0236.34006MR229912
  23. MILLIONŠČIKOV, V., Typicality of almost reducible systems with almost periodic coefficients, Differential Equations, 14 (1978), 448-450. Zbl0434.34027MR508462
  24. MOSER, J., An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helvetici, 56 (1981), 198-224. Zbl0477.34018MR630951
  25. MOSER, J.- PÖSCHEL, J., An extension of a result by Dinaburg and Sinai on quasi-periodic potentials, Comment. Math. Helvetici, 59 (1984), 39-85. Zbl0533.34023MR743943
  26. NERURKAR, M., Positive exponents for a dense class of continuous S ⁢ L ⁢ 2 , R -valued cocycles which arise as solutions to strongly accessible linear differential systems, Contemp. Math., vol. 215 (1998), AMS, 265-278. Zbl0953.34041MR1603050
  27. PALMER, K., Exponential dichotomies and transversal homoclinic points, Jour. of Diff. Eqns., 55 (1984), 225-256. Zbl0508.58035MR764125
  28. PALMER, K., Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, Vol. 1 (1988), 265-306. Zbl0676.58025MR945967
  29. SACKER, R. J.- SELL, G. R., A spectral theory for linear differential systems, Journal of Differential Equations, 27 (1978), 320-358. Zbl0372.34027MR501182
  30. SINAI, YA., Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential, Journal of Statistical Physics, 46 (1987), 861-909. Zbl0682.34023MR893122
  31. WOJTKOWSKI, M., Invariant families of cones and Lyapunov exponents, Ergod. Th. & Dynam. Sys., 5 (1985), 145-161. Zbl0578.58033
  32. WOJTKOWKI, M., Principles for the Design of Billiards with Nonvanishing Lyapunov Exponents, Comm. Math. Phys., 105 (1986), 391-414. Zbl0602.58029MR848647
  33. YOUNG, L. S., Some open sets of nonuniformly hyperbolic cocycles, Ergod. Th. & Dynam. Sys., 13 (1993), 409-415. Zbl0797.58041
  34. YOUNG, L. S., Lyapounov exponents for some quasi-periodic cocycles, Ergod. Th. & Dynam. Sys., 17 (1997), 483-504. Zbl0873.28013

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.