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Zero measure spectrum for the almost Mathieu operator. (English) Zbl 0814.11040

The almost Mathieu (or Harper’s) operator on \(\ell^ 2(\mathbb{Z})\) is given by \[ (H_{\alpha,\lambda,\theta}u)(n)= u(n + 1) + u(n -1) + \lambda \cos (2\pi\alpha n + \theta) u(n) \] and is related to Schrödinger’s operator and to the rotation number. Upper and lower estimates for the Lebesgue measure of the spectrum of the operator are obtained for each \(\lambda\), \(\theta\), \(\alpha\). The measure is deduced to be \(| 4- | \lambda||\) for all \(\lambda\), \(\theta\) and for \(\alpha\) with unbounded partial quotients, thus for all \(\lambda\), \(\theta\) and for almost all \(\alpha\). When \(| \lambda| = 2\), the spectrum is a Cantor set of measure 0. In an interesting mix of operator and number theory, it is further shown that the Hausdorff dimension of the spectrum is at most 1/2 for irrational \(\alpha\) such that for any \(c > 0\), there are infinitely many rationals \(p_ n/q_ n\) satisfying \[ \left | \alpha - {p_ n\over q_ n}\right | < {c\over q_ n^ 4}, \] a set of dimension 1/2 (for \(\alpha\) with unbounded quotients, the denominator on the RHS is \(q^ 2_ n)\).

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
34L05 General spectral theory of ordinary differential operators
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H99 Hamiltonian and Lagrangian mechanics
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[1] Aubry, S., Andre, G.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc.3, 133–164 (1980) · Zbl 0943.82510
[2] Avron, J., Simon, B.: Almost periodic Schrödinger operators. II. The integrated density of states. Duke Math. J.50, 369–391 (1983) · Zbl 0544.35030
[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) · Zbl 0724.47002
[4] Bellissard, J., Lima, R., Testard, D.: A metal-insulator transition for the almost Mathieu model. Commun. Math. Phys.88, 207–234 (1983) · Zbl 0542.35059
[5] Bellissard, J., Simon, B.: Cantor spectrum for the almost Mathieu equation. J. Funct. Anal.48, 408–419 (1982) · Zbl 0516.47018
[6] Chambers, W.: Linear network model for magnetic breakdown in two dimensions. Phys. Rev. A140, 135–143 (1965)
[7] Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math.99, 225–246 (1990) · Zbl 0665.46051
[8] Chulaevsky, V., Delyon, F.: Purely absolutely continuous spectrum for almost Mathieu operators. J. Stat. Phys.55, 1279–1284 (1989) · Zbl 0714.34129
[9] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin, Heidelberg, New York: Springer 1987
[10] Delyon, F.: Absence of localization for the almost Mathieu equation. J. Phys. A20, L21-L23 (1987) · Zbl 0622.34024
[11] Harper, P.G.: Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. Lond. A68, 874–892 (1955) · Zbl 0065.23708
[12] Helffer, B., Sjostrand, J.: Semi-classical analysis for Harper’s equation. III. Cantor structure of the spectrum. Mém. Soc. Math. France (N.S.)39, 1–139 (1989)
[13] Hofstadter, D.R.: Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field. Phys. Rev. B14, 2239–2249 (1976)
[14] Falconer, K.J.: The geometry of fractal sets. Cambridge: Cambridge University Press 1985 · Zbl 0587.28004
[15] Fröhlich, J., Spencer, T., Wittwer, P.: Localization for a class of one dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys.132, 5–25 (1990) · Zbl 0722.34070
[16] Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers, Fifth ed. Oxford: Oxford University Press, 1979 · Zbl 0423.10001
[17] Last, Y.: A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Commun. Math. Phys.151, 183–192 (1993) · Zbl 0782.34084
[18] Last, Y., Wilkinson, M.: A sum rule for the dispersion relations of the rational Harper’s equation. J. Phys. A25, 6123–6133 (1992) · Zbl 0772.35059
[19] Simon, B.: Almost periodic Schrödinger operators: a review. Adv. Appl. Math.3, 463–490 (1982) · Zbl 0545.34023
[20] Sinai, Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Stat. Phys.46, 861–909 (1987) · Zbl 0682.34023
[21] Thouless, D.J.: Bandwidth for a quasiperiodic tight binding model. Phys. Rev. B28, 4272–4276 (1983)
[22] Thouless, D.J.: Scaling for the discrete Mathieu equation. Commun. Math. Phys.127, 187–193 (1990) · Zbl 0692.34021
[23] Thouless, D.J., Tan, Y.: Total bandwidth for the Harper equation. III. Corrections to scaling. J. Phys. A24, 4055–4066 (1991)
[24] Thouless, D.J., Tan, Y.: Scaling, localization and bandwidths for equations with competing periods. Physica A177, 567–577 (1991)
[25] Toda, M.: Theory of nonlinear lattices, 2nd Ed., Chap. 4. Berlin, Heidelberg, New York: Springer 1989 · Zbl 0694.70001
[26] Watson, G.I.: WKB analysis of energy band structure of modulated systems. J. Phys. A24, 4999–5010 (1991)
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