Cantor Spectrum Research Articles

A condition of Boshernitzan and uniform convergence in the multiplicative ergodic theorem

This article is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan [6], [7], every locally constant SL(2,R)-valued cocycle is uniform. As a consequence, the corresponding Schrodinger operators exhibit Cantor spectrum of Lebesgue measure zero

Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials

We consider discrete one-dimensional Schrödinger operators whose potentials belong to minimal subshifts of low combinatorial complexity and prove for a large class of such operators that the spectrum is a Cantor set of zero Lebesgue measure. This is obtained through an analysis of the frequencies of the subwords occurring in the potential. Our results cover most circle map and Arnoux–Rauzy potentials.

A nonperturbative Eliasson's reducibility theorem

This paper is concerned with discrete, one-dimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasi-periodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated first-order system, a quasi-periodic skew-product, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schrödinger operators. Finally we prove that Cantor spectrum implies the existence of a Gδ-set of energies whose Schrödinger cocycle is not reducible to constant coefficients.

Positive measure spectrum for Schrödinger operators with periodic magnetic fields

We study Schrödinger operators with periodic magnetic field in R2, in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that, even without electric potential, the spectrum has positive measure if the magnetic field is a perturbation of a constant one.

Uniform Cantor Singular Continuous Spectrum for Nonprimitive Schrödinger Operators

It is shown that some Schrodinger operators, with nonprimitive substitution potentials, have pure singular continuous Cantor spectrum with null Lebesgue measure for all elements in the respective hulls.

The spectrum of a class of almost periodic operators

For almost Mathieu operators, it is shown that the occurrence of Cantor spectrum and the existence, for every point in the spectrum and suitable phase parameters, of at least one localized eigenfunction which decays exponentially are inconsistent properties.

Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals

The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author.

Discrete and Cantor Spectrum for Neumann Laplacians of Combs

AbstractExtending results of [HSS] on the construction of Neumann Laplacians of combs with given essential spectrum S = S [0,∞], we show that, in addition to the essential spectrum, a bounded sequence of discrete eigenvalues can be prescribed. More generally, we find that certain types of spectra can be constructed precisely, in a bounded interval [0, s].

Electronic properties of 2D quasicrystals: level spacing distribution and diffusion

We present numerical calculations of the band spectrum of periodic approximants of the octagonal quasiperiodic tiling. We have studied a one-parameter family of Hamiltonians including the pure hopping case, the Laplacian and a regime where atomic potentials prevail. Working on approximants allows us to avoid boundary effects and to identify properly the nature of the spectrum. We have found a range of the parameter for which the spectrum consists in only one band. Moreover, the Lebesgue measure is finite except maybe in a small domain of the parameter. We have also studied the statistics of levels. We have obtained a transition between a regime of level repulsion (quantum chaos) and another of level clustering (Cantor spectrum). As a result, in the strong coupling regime the integrated density of spacings behaves like P( s) ∼ s − α , s → 0. We have related α to the quantum diffusion exponent of a wave packet.

Absence of Cantor spectrum for a class of Schrödinger operators

It is shown that the complete localization of eigenvectors for the almost Mathieu operator entails the absence of Cantor spectrum for this operator.

Metamorphosis of a Cantor spectrum due to classical chaos.

We study how a Cantor spectrum, its level statistics, and corresponding dynamics are affected by the onset of classical chaos. While the spectrum undergoes visible changes, its level spacing distribution follows an inverse power law p(s)\ensuremath{\sim}${\mathit{s}}^{\mathrm{\ensuremath{-}}3/2}$ on small scales. We find a crossover which is manifested in the time domain by two diffusive regimes characterized by a classical and a quantum-mechanical diffusion coefficient. In the strong quantum limit we show by means of a transformation that the spectrum is governed by the IintegrableP Harper equation, even if the classical phase space is strongly chaotic.

Band spectrum of the octagonal quasicrystal: Finite measure, gaps, and chaos.

In this paper, we present numerical calculations of the band spectrum of periodic approximants of the octagonal quasiperiodic tiling. We have studied a one-parameter family of Hamiltonians, including the pure hopping case, the Laplacian, and a regime in which atomic potentials prevail. We have found a parameter range over which the spectrum consists of only one band. Moreover, the Lebesgue measure is finite except maybe in a small domain of the parameter range. We have also studied the statistics of the levels. We have obtained a transition between a regime of level repulsion (quantum chaos) and another of level clustering (Cantor spectrum).

Quasiperiodicity and long-range order in a magnetic system.

We study the 1D quasiperiodic quantum Ising model in a transverse field, which exhibits long-range order above a critical coupling and describes the phase diagram of the 2D classical Ising-model quasiperiodic in one direction. We numerically determine the correlation length, magnetization, and energy spectrum of this system. The site dependency of the magnetization is described by a new order parameter, the width of the magnetization. The scalings delta and f(..cap alpha..) associated with the Cantor spectrum are found to exhibit special features at the onset of long-range order.

Remarks on linear differential systems with measurable coefficients

We discuss several topics having to do with linear differential equations with locally integrable coefficients. These include gap labelling and Cantor spectrum for the random Schrodinger operator, and Coppel’s perturbation theorem.

Cantor spectrum for an almost periodic Schrödinger equation and a dynamical map

The spectrum of an incommensurate electron system is studied through a dynamical map. The imbedded horseshoe structure in the map leads to the infinite hierarchical structure of the energy spectrum. A partial renormalization group analysis of the map develops the scaling for the spectrum numerically found previously.

Cantor spectrum for the almost Mathieu equation

For a dense G δ of pairs (λ, α) in R 2, we prove that the operator ( Hu)( n) = u( n + 1) + u( n −1) + λ cos(2 παn + θ) u( n) has a nowhere dense spectrum. Along the way we prove several interesting results about the case α = p q of which we mention: (a) If qθ is not an integral multiple of π, then all gaps are open, and (b) If q is even and θ is chosen suitably, then the middle gap is closed for all λ.

Chaotic States of Almost Periodic Schrödinger Operators

A one-dimensional quantum mechanical model defined by the quadratic mapping Hamiltonian which is almost periodic is presented. The model admits a Cantor spectrum of Lebesgue measure zero with a singular continuous measure and produces extended states displaying an unexpected chaotic behavior at large distances.