Spectral properties of one dimensional quasi-crystals

AbstractIn this paper we prove that the one dimensional Schrödinger operator onl2(ℤ) with potential given by: $$\upsilon (n) = \lambda \chi _{[1 - \alpha , 1[} (x + n\alpha )\alpha \notin \mathbb{Q}$$ has a Cantor spectrum of zero Lebesgue measure for any irrationalα and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepGe9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhaiiaacq% WFiiIZtuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbbiab% +nj8ubaa!4628!\[x \in \mathbb{T}$$ .

[1]  J. Sethna,et al.  Universal Transition from Quasiperiodicity to Chaos in Dissipative Systems , 1982 .

[2]  P J Steinhardt,et al.  The physics of quasicrystals , 1987 .

[3]  Band spectrum for an electron on a Sierpinski gasket in a magnetic field , 1987 .

[4]  J. Luck,et al.  Phonon spectra in one-dimensional quasicrystals , 1986 .

[5]  Chao Tang,et al.  Localization Problem in One Dimension: Mapping and Escape , 1983 .

[6]  F. Martinelli,et al.  Introduction to the mathematical theory of Anderson localization , 1986 .

[7]  Banavar,et al.  Quasiperiodic lattice: Electronic properties, phonon properties, and diffusion. , 1986, Physical review. B, Condensed matter.

[8]  M. K. Ali,et al.  Scaling and eigenstates for a class of one-dimensional quasiperiodic lattices , 1988 .

[9]  C. Janot,et al.  Quasicrystalline materials : proceedings of the I.L.L./CODEST Workshop, Grenoble 21-25 March 1988 , 1988 .

[10]  B. Simon.  Almost periodic Schrödinger operators: A Review , 1982 .

[11]  M. Casdagli.  Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation , 1986 .

[12]  András Sütő,et al.  Singular continuous spectrum on a cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian , 1989 .

[13]  P. Bougerol,et al.  Products of Random Matrices with Applications to Schrödinger Operators , 1985 .

[14]  JACOBI MATRICES WITH RANDOM POTENTIALS TAKING FINITELY MANY VALUES , 1989 .

[15]  Kohmoto,et al.  Resistance of a one-dimensional quasicrystal: Power-law growth. , 1987, Physical review. B, Condensed matter.

[16]  F. Martinelli,et al.  Multiple tunnelings in d-dimensions : a quantum particle in a hierarchical potential , 1985 .

[17]  J. Cassels,et al.  An Introduction to Diophantine Approximation , 1957 .

[18]  J. Cahn,et al.  Metallic Phase with Long-Range Orientational Order and No Translational Symmetry , 1984 .

[19]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[20]  H. Kunz,et al.  Cantor spectrum and singular continuity for a hierarchical Hamiltonian , 1989 .

[21]  Dimitri Petritis,et al.  Absence of localization in a class of Schrödinger operators with quasiperiodic potential , 1986 .

[22]  On the Energy Transport in Linear Chains , 1974 .

[23]  Seunghwan Kim,et al.  Renormalization of Quasiperiodic Mappings , 1985 .

[24]  Ali,et al.  Dynamical maps, Cantor spectra, and localization for Fibonacci and related quasiperiodic lattices. , 1988, Physical review letters.

[25]  M. Kohmoto.  Metal-Insulator Transition and Scaling for Incommensurate Systems , 1983 .

[26]  M. Barnsley,et al.  Almost periodic Jacobi matrices associated with Julia sets for polynomials , 1985 .

[27]  András Sütő,et al.  The spectrum of a quasiperiodic Schrödinger operator , 1987 .

[28]  R. Rammal,et al.  Spectrum of harmonic excitations on fractals , 1984 .

[29]  E. Scoppola,et al.  The density of states for almost periodic Schrödinger operators and the frequency module: A counter-example , 1982 .

[30]  J. Luck,et al.  Cantor spectra and scaling of gap widths in deterministic aperiodic systems. , 1989, Physical review. B, Condensed matter.

[31]  B A Huberman,et al.  Ultradiffusion: the relaxation of hierarchical systems , 1985 .

[32]  D. Bessis,et al.  Chaotic States of Almost Periodic Schrödinger Operators , 1982 .

[33]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[34]  David A. Rand,et al.  One-dimensional schrodinger equation with an almost periodic potential , 1983 .

[35]  F. Axel,et al.  VIBRATIONAL MODES IN A ONE DIMENSIONAL "QUASI-ALLOY" : THE MORSE CASE , 1986 .