×

zbMATH — the first resource for mathematics

Spectral properties of one dimensional quasi-crystals. (English) Zbl 0825.58010
Summary: We prove that the one-dimensional Schrödinger operator on \(l^ 2(\mathbb{Z})\) with potential given by: \[ v(n)= \lambda_{\chi[1- \alpha,1[}(x+ n\alpha),\qquad \alpha\not\in\mathbb{Q} \] has a Cantor spectrum of zero Lebesgue measure for any irrational \(\alpha\) and any \(\lambda> 0\). We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all \(x\in \mathbb{T}\).

MSC:
37E99 Low-dimensional dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
82D25 Statistical mechanical studies of crystals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexander, S.: Some properties of the spectrum on the Sierpinsky gasket in a magnetic field. Phys. Rev. B29, 5504-5508 (1984) · doi:10.1103/PhysRevB.29.5504
[2] Aubry, S., Andre, G.: Analyticity breaking and the Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc.3, 133-164 (1980) · Zbl 0943.82510
[3] Axel, F., Allouche, J.P., Kleman, M., Mendes-France, M., Peyriere, J.: Vibrational modes in a one dimensional ?quasi-alloy?: the Morse case. J. Phs. C3,47, C3 181-C3 186 (1986) · Zbl 0696.10050
[4] Bransley, M.F., Geronimo, J.S., Harrington, A.N.: Almost periodic Jacobi matrices associated with Julia sets for polynomials. Commun. Math. Phys.99, 303-317 (1985) · Zbl 0574.58016 · doi:10.1007/BF01240350
[5] Bellissard, J.: Almost periodicity in solid state physics andc* algebras, H. Bohr Centennary Conference on almost periodic functions, to appear (1987)
[6] Bellissard, J., Bessis, D., Moussa, P.: Chaotic states of almost periodic Schrödinger operators. Phys. Rev. Lett.49, 701-704 (1982) · doi:10.1103/PhysRevLett.49.701
[7] Bellissard, J., Scoppola, E.: The density of states for almost periodic Schrödinger operators and the frequency module: a counterexample. Commun. Math. Phys.85, 301-308 (1982) · Zbl 0505.46056 · doi:10.1007/BF01254461
[8] Bougerol, Ph., Lacroix, J.: Products of random matrices with applications to Schrödinger operators. Boston, Stuttgart: Birkhäuser · Zbl 0572.60001
[9] Casdagli, M.: Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation. Commun. Math. Phys.107, 295 (1986) · Zbl 0606.39004 · doi:10.1007/BF01209396
[10] Delyon, F., Petritis, D.: Absence of localization in a class of Schrödinger operators with quasiperiodic potential. Commun. Math. Phys.103, 441 (1986) · Zbl 0604.35072 · doi:10.1007/BF01211759
[11] Ghez, J.M., Wang, W., Rammal, R., Pannetier, B., Bellissard, J.: Band spectrum for an electron on a Serpinsky gasket in a magnetic field. Sol. State Commun.64, 1291-1294 (1987) · doi:10.1016/0038-1098(87)90628-4
[12] Gumbs, G., Ali, M.K.: Scaling and eigenstates for a class of one dimensional quasi-periodic lattices. J. Phys. A21, L 517-L 521 (1988) · doi:10.1088/0305-4470/21/9/007
[13] Gumbs, G., Ali, M.K.: Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices. Phys. Rev. Lett.60, 1081-1084 (1988) · doi:10.1103/PhysRevLett.60.1081
[14] Huberman, B.A., Kerzberg, M.: Ultra-diffusion: the relaxation of hierarchical systems. J. Phys. A18, L 331-L 336 (1985) · doi:10.1088/0305-4470/18/6/013
[15] Janot, Ch., Dubois, J.M.: Editors: Quasicrystalline materials. Grenoble 21-25 march 1988. Singapore: World Scientific 1988
[16] Jona-Lasinio, G., Martinelli, F., Scoppola, E.: Multiple tunneling ind-dimensions: a quantum particle in a hierarchical potential. Ann. Inst. Henri Poincaré42, 73-108 (1985) · Zbl 0586.35030
[17] Kadanoff, L.P., Kohmoto, M., Tang, C.: Localization problem in one dimension: mapping and escape. Phys. Rev. Lett.50, 1870-1872 (1983) · doi:10.1103/PhysRevLett.50.1870
[18] Kalugin, P.A., Kilaev, A.Yu., Levitov, S.: Electron spectrum of a one dimensional quasi-crystal. Sov. Phys. JETP64, 410-415 (1986)
[19] Komoto, M.: Metal insulator transition and scaling for incommensurate system. Phys. Rev. Lett.51, 1198-1201 (1983) · doi:10.1103/PhysRevLett.51.1198
[20] Komoto, M., Banavar, J.R.: Quasi-periodic lattice: electronic properties and diffusion, Phys. B34, 563-566 (1986)
[21] Kotani, S.: Jacobi matrices with random potentials taking finitely many values, Preprint Tokyo (1989) · Zbl 0713.60074
[22] Kunz, H., Livi, R., Suto, A.: Cantor spectrum and singular continuity for a hierarchical hamiltonian. Commun. Math. Phys.122, 643-679 (1989) · Zbl 0687.35062 · doi:10.1007/BF01256499
[23] Lang, S.: Introduction to diophantine approximations, Reading MA; Addison-Wesley, 1966 · Zbl 0144.04005
[24] Levitov, L.S.: Renormalization group for a quasiperiodic Schrödinger operator. J. Stat. Phys. (to appear)
[25] Luck, J.M.: Cantor spectra and scaling of gap widths in deterministic aperiodic systems. Phys. Rev. (to appear)
[26] Luck, J.M., Petritis, D.: Phonon spectra in one-dimensional quasicrystal. J. Stat. Phys.42, 289-310 (1986) · doi:10.1007/BF01127714
[27] Machida, K., Nakano, M.: Soliton lattice structure and mid-gap band in nearly commensurate charge-density-wave states. II Self-similar band structure and coupling constant dependence. Phys. Rev. B34, 5073-5081 (1986) · doi:10.1103/PhysRevB.34.5073
[28] Martinelli, F., Scoppola, E.: Introduction to the mathematical theory of Anderson localization. Rivista del Nuovo Cimento10 (1987)
[29] Ostlund, S., Kim, S.: Renormalization of quasi periodic mappings. Physica Scripta9, 193-198 (1985) · Zbl 1063.37526 · doi:10.1088/0031-8949/1985/T9/031
[30] Ostlund, S., Prandit, R., Rand, D., Schnellnhuber, H.J., Siggia, E.D.: One dimensional Schrödinger equation with an almost periodic potential. Phys. Rev. Lett.50, 1873-1877 (1983) · doi:10.1103/PhysRevLett.50.1873
[31] Rammal, R.: Spectrum of harmonic excitations on fractals. J. Phys.45, 191-206 (1984)
[32] Rammal, R., Lubensky, T.C., Toulouse, G.: Supraconducting networks in a magnetic field. Phys. Rev. B27, 2820-2829 (1983) · doi:10.1103/PhysRevB.27.2820
[33] Rand, D., Ostlund, S., Sethna, J., Siggia, E.D.: Universal transition from quasi periodicity to chaos in dissipative systems. Phys. Rev. Lett.49, 132-135 (1982) · Zbl 0538.58025 · doi:10.1103/PhysRevLett.49.132
[34] Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York: Academic Press 1972 · Zbl 0242.46001
[35] Schechtman, D., Blech, I., Gratias, D., Cahn, J.V.: Metallic phase with long range orientational order and no translational symmetry. Phys. Rev. Lett.53, 1951-1953 (1984) · doi:10.1103/PhysRevLett.53.1951
[36] Simon, B.: Almost periodic Schrödinger operators: A review. Adv. Appl. Math.3, 463-490 (1982) · Zbl 0545.34023 · doi:10.1016/S0196-8858(82)80018-3
[37] Steinhardt, P.J., Ostlund, S.: The physics of quasicrystals. Singapore: World Scientific 1987 · Zbl 0997.82516
[38] Sutherland, B., Kohmoto, M.: Resistance of a one dimensional quasi crystal: power law growth. Phys. Rev. B36, 5877-5886 (1987) · doi:10.1103/PhysRevB.36.5877
[39] Süto, A.: The spectrum of a quasi-periodic Schrödinger operator. Commun. Math. Phys.111, 409-415 (1987) · Zbl 0624.34017 · doi:10.1007/BF01238906
[40] Süto, A.: Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian. J. Stat. Phys. (to appear)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.