Positive Lyapunov exponents for Schrödinger operators with quasi- periodic potentials. (English) Zbl 0745.34046

The authors present a new, simple way to estimate the Lyapunov exponent of solutions of the finite-difference Schrödinger equation, \[ ((H- E)\psi)(n){\buildrel{\text{def}}\over =}-[\psi(n+1)+\psi(n-1)]+[\lambda f(\alpha n+\theta)]\psi(n), \] where \(f\) is a non-constant real-analytic function of period 1 and \(\alpha\) is irrational. For \(\lambda\) large they prove that the Lyapunov exponent is positive for every energy \(E\) in the spectrum of \(H\) and a.e. \(\theta\). The main idea of the approach is to deform the phase \(\theta\) to the complex plane, and then to recover the information for real \(\theta\) using an elementary extension of Jensen’s formula.


34D08 Characteristic and Lyapunov exponents of ordinary differential equations
39A11 Stability of difference equations (MSC2000)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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[1] Albanese, C.: Quasiperiodic Schrödinger Operators with Pure Absolutely Continuous Spectrum. Preprint, CIMS
[2] Aubry, S.: Solid State Sci.8, 264 (1978)
[3] Chulaevsky, V., Delyon, F.: Purely absolutely continuous spectrum for the almost Mathieu operators. J. Stat. Phys.55, 1279–1284 (1989) · Zbl 0714.34129
[4] Chulaevsky, V., Sinai, Ya.: Anderson Localization for multi-frequency quasi-periodic potentials in one dimension. Commun. Math. Phys.125, 91–112 (1989) · Zbl 0743.60058
[5] Craig, W., Simon, B.: Subharmonicity of the Liapunov index. Duke Math. J.50, 551–560 (1983) · Zbl 0518.35027
[6] Delyon, F.: J. Phys. A20, L21 (1987) · Zbl 0622.34024
[7] Dinaburg, E., Sinai, Ya.: The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl.9, 279–289 (1975) · Zbl 0333.34014
[8] Eliasson, L.: Floquet Solutions for the One-Dimensional Quasi-Periodic Schrödinger Equation. Preprint, University of Stockholm · Zbl 0753.34055
[9] Gordon, A.: Usp. Math. Nauk.31, 257 (1976) [in Russian]
[10] Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one-dimensional quasiperiodic potentials. Commun. Math. Phys.132 5–25 (1990) · Zbl 0722.34070
[11] Herman, M.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le charactère local d’un théorème d’Arnold et de Moser sur le tore en dimension 2. Commun. Math. Helv.58, 453–502 (1983) · Zbl 0554.58034
[12] Hiramoto, H., Kohmoto, M.: Phys. Rev. Lett.62, 2714 (1989)
[13] Kunz, H., Souillard, B.: Commun. Math. Phys.78, 201–246 (1980) · Zbl 0449.60048
[14] Kingman, J. F. C.: Subadditive Processes, Lecture Notes in Mathematics, vol.539. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0367.60030
[15] Olver, F. W. J.: Asymptotics and Special Functions. New York: Academic Press 1974 · Zbl 0303.41035
[16] Pastur, L.: Spectral Properties of disordered systems in one-body approximation. Commun. Math. Phys.75, 179 (1980) · Zbl 0429.60099
[17] Reed, M., Simon, B.: Methods of modern mathematical physics, vol. 1–4. New York: Academic Press 1982 · Zbl 0517.47006
[18] Sarnak, P.: Spectral behavior of Quasi-periodic potentials. Commun. Math. Phys.84, 377–401 (1982) · Zbl 0506.35074
[19] Sinai, Ya. G.: Anderson localization for one-dimensional difference Schrödinger operator with quasi-periodic potential. J. Stat. Phys.46, 861–918 (1987) · Zbl 0682.34023
[20] Cycon, H.L. et al.: Schrödinger Operators. Berlin, Heidelberg, New York: Springer 1987 · Zbl 0619.47005
[21] Spencer, T.: The Schrödinger equation with a random potential–A mathematical review. In: Critical Phenomena, Random Systems, Gauge Theories, Les Houches, XLIII. d Osterwalder K., Stora, R. (eds.) · Zbl 0655.60050
[22] Surace, S.: Trans. Am. Math. Soc.320, 321 (1990) · Zbl 0712.34094
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