A metal-insulator transition for the almost Mathieu model. (English) Zbl 0542.35059

The authors examine the spectral properties of the almost Mathieu Hamiltonian \[ H_ x^{\mu}\psi(n)=\psi(n+1)+\psi(n-1)+2\mu \quad \cos(x-n\theta)\psi(n),\quad n\in {\mathbb{Z}} \] where \(\theta\) is an irrational number and \(x\in T\). For sufficiently small coupling constant \(\mu\) and any x they show that there is a closed energy set of non-zero measure in the absolutely continuous spectrum of H. As well for sufficiently high \(\mu\) and almost all x the existence is established of a set of eigenvalues whose closure has positive measure. These results are obtained for a subset of irrational numbers \(\theta\) of full Lebesgue measure.
The first result depends on a reformulation of Rüssmann of results of Dinaberg and Sinai for studying the spectrum of \(H_ x\) for small values of \(\mu\) and has its basis in Newton’s algorithm for computing roots of certain equations together with a small divisor analysis. For the second result the authors use André-Aubry duality to show the existence of an infinite number of eigenvalues with exponentially localised eigenfunctions and finally techniques of number theory to show that the closure of this set is strictly positive.
Reviewer: M.Thompson


35P25 Scattering theory for PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
47A40 Scattering theory of linear operators
Full Text: DOI


[1] Anderson, P.N.: Absence of diffusion in certain random lattices. Phys. Rev.109, 1492 (1958)
[2] André, G., Aubry, S.: Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc.3, 133 (1980) · Zbl 0943.82510
[3] Arnold, V.I.: Small divisor problems in classical and celestial mechanics. Usp. Mat. Nauk.18, no 6 (114), 91-192 (1963)
[4] Avron, J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Am. Math. Soc.6, 81-86 (1982) · Zbl 0491.47014
[5] Azbell, M.Ya.: Energy spectrum of a conducting electron in a magnetic field. Zh. Eksp. Teor. Fiz.46, 929 (1964) and Sov. Phys. JETP19, 634 (1964)
[6] Bellissard, J., Testard, D.: Almost periodic Hamiltonian: an algebraic approach. Marseille Preprint (1981) · Zbl 0499.46041
[7] Schultz, H.J., Jerome, D., Mazaud, A., Ribault, M., Bechgaard, K.: Possibility of superconducting precursor effects in quasi one-dimensional organic conductors: theory and experiments. J. Phys. (France)42, no 7, 991-1002 (1981)
[8] Bloch, A.N.: Chemical trends in organic conductors: stabilization of the nearly one-dimensional metallic state. In: Organic conductors and semi-conductors. Proceeding 1976, Pal, L., Grüner, G., Jánossy, A., Sólyon, J. (eds.). Lecture Notes in Physics, Vol. 65, p. 317. Berlin, Heidelberg, New York: Springer 1977
[9] Dinaburg, E.I., Sinai, Ya.G.: The one dimensional Schrödinger equation with a quasi periodic potential. Funct. Anal. Appl.9, 279 (1976) · Zbl 0333.34014
[10] Falicov, L.M., Hsu, W.Y.: Level quantization and broadening for band electrons in a magnetic field: magneto-optics throughout the band. Phys. Rev. B13, 1595 (1976)
[11] Fröhlich, H.: On the theory of superconductivity: the one-dimensional case. Proc. R. Soc. A223, 296 (1954) · Zbl 0055.44105
[12] Gordon, A.Ya.: The point spectrum of the one-dimensional Schrödinger operator. Usp. Mat. Nauk. (Russian)31, no 4 (190), 257 (1976)
[13] Halmos, P.R.: Lectures on ergodic theory. New York: Chelsea Publ. Company 1961 · Zbl 0161.11401
[14] Harper, P.G.: Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A68, 874-892 (1955) · Zbl 0065.23708
[15] Hermann, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. IHES no 49, 5-234 (1979)
[16] Hermann, M.R.: Une méthode pour minorer les exposants de Lyapounov. Preprint (1982), Ecole Polytechnique, Palaiseau (France)
[17] Hofstadter, D.R.: Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B14, 2239 (1976)
[18] Kolmogorov, A.N.: On conservation of conditionnally periodic motion for a small change in Hamilton’s function. Dokl. Akad. Nauk. SSSR (Russian)98, 527 (1954)
[19] Kolmogorov, A.N.: Théorie générale des systèmes dynamiques et mécanique classique. Proc. Int. Congress of Math. Amsterdam (1957)
[20] Little, W.A.: Possibility of synthesizing an organic superconductor. Phys. Rev. A134, 1416 (1964)
[21] Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Ann.169, 136-176 (1967) · Zbl 0149.29903
[22] Peierls, R.S.: Quantum theory of solids, Chap. VII. Oxford: Pergamon Press 1955 and Z. Phys.80, 763 (1933)
[23] Rauh, A.: Degeneracy of Landau levels in crystals. Phys. Stat. Solidi B65, K 131 (1974); B69, K 9 (1975)
[24] Rüssmann, H.: On the one dimensional Schrödinger equation with a quasi-periodic potential. Ann. N. Y. Acad. Sci.357, 90 (1980)
[25] Rüssmann, H.: Note on sums containing small divisors. Commun. Pure Appl. Math.29, 755 (1976) · Zbl 0336.35020
[26] Rüssmann, H.: On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. In: Lecture Notes in Physics, Vol. 38, p. 598. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0319.35017
[27] Sarnak, P.: Spectral behaviour of quasi-periodic potential. Commun. Math. Phys.84, 377 (1982) · Zbl 0506.35074
[28] Titchmarsh, E.C.: Eigenfunctions expansions associated with second order differential equations, Vol. 1. Oxford: Oxford University Press 1958 · Zbl 0097.27601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.