Sturmian Potentials Research Articles

Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators

We introduce a notion of \beta -almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded \beta -almost periodic potentials. Applications include the first sharp arithmetic spectral criterion for the entire family of supercritical analytic quasiperiodic Schrödinger operators and arithmetic spectral/quantum dynamical criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator. In particular, we disprove a 1994 conjecture of Wilkinson–Austin.

Invariants of the trace map and uniform spectral properties for discrete Sturmian Dirac operators

We establish invariants for the trace map associated to a family of 1D discrete Dirac operators with Sturmian potentials. Using these invariants we prove that the operators have purely singular continuous spectrum of zero Lebesgue measure, uniformly on the mass and parameters that define the potentials. For rotation numbers of bounded density we prove that these Dirac operators have purely $\alpha$-continuous spectrum, as to the Schrodinger case, for some $\alpha \in (0,1)$. To the Sturmian Schrodinger and Dirac models we establish a comparison between invariants of the trace maps, which allows to compare the numbers $\alpha$'s and lower bounds on transport exponents.

On the spectral Hausdorff dimension of 1D discrete Schrödinger operators under power decaying perturbations

We show that spectral Hausdorff dimensional properties of discrete Schrodinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component.

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

The Feshbach–Villars equations, like the Klein–Gordon equation, are relativistic quantum mechanical equations for spin-0 particles.We write the Feshbach–Villars equations into an integral equation form and solve them by applying the Coulomb–Sturmian potential separable expansion method. We consider boundstate problems in a Coulomb plus short range potential. The corresponding Feshbach–Villars CoulombGreen’s operator is represented by a matrix continued fraction.

Transport exponents of Sturmian Hamiltonians

We consider discrete Schr\"odinger operators with Sturmian potentials and study the transport exponents associated with them. Under suitable assumptions on the frequency, we establish upper and lower bounds for the upper transport exponents. As an application of these bounds, we identify the large coupling asymptotics of the upper transport exponents for frequencies of constant type. We also bound the large coupling asymptotics uniformly from above for Lebesgue-typical frequency. A particular consequence of these results is that for most frequencies of constant type, transport is faster than for Lebesgue almost every frequency. We also show quasi-ballistic transport for all coupling constants, generic frequencies, and suitable phases.

Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians

We consider the spectrum of discrete Schrödinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.

Gibbs-like measure for spectrum of a class of quasi-crystals

AbstractLet α∈(0,1) be an irrational, and [0;a1,a2,…] the continued fraction expansion of α. Let Hα,V be the one-dimensional Schrödinger operator with Sturmian potential of frequency α. Suppose the potential strength V >20 and the sequence (ai)i≥1 is bounded. We proceed by developing some new ideas on dimensional theory of Cookie-cutter sets. We prove that the spectral generating bands satisfy the principles of bounded variation and bounded covariation, and then we show that there exists a Gibbs-like measure on the spectrum σ(Hα,V). As an application, we prove that where s* and s* are the lower and upper pre-dimensions. Moreover, if (an)n≥1 is ultimately periodic, then s* =s*.

DYNAMICAL BOUNDS FOR STURMIAN SCHRÖDINGER OPERATORS

The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a quasiperiodical Sturmian potential with respect to the golden mean has been investigated intensively in recent years. Damanik and Tcheremchantsev developed a method in [10] and used it to exhibit a non trivial dynamical upper bound for this model. In this paper, we use this method to generalize to a large family of Sturmian operators dynamical upper bounds and show at sufficently large coupling anomalous transport for operators associated to irrational number with a generic diophantine condition. As a counterexample, we exhibit a pathological irrational number which does not verify this condition and show its associated dynamic exponent only has ballistic bound. Moreover, we establish a global lower bound for the lower box counting dimension of the spectrum that is used to obtain a dynamical lower bound for bounded density irrational numbers.

Dynamical bounds for Sturmian Schrödinger operators

The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a Sturmian potential with respect to the golden number has been investigated intensively in recent years. Damanik and Tcheremchantsev developed a method and found a non-trivial dynamical upper bound for transport exponents for this model. This method can be generalized to obtain results for almost all irrational numbers. As a counter example, we exhibit a pathological irrational number with no possible better bound. Moreover, we establish a global lower bound for the lower box dimension of the spectrum that could be used to obtain a dynamical lower bound for irrational numbers with bounded density. To cite this article: L. Marin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials

Damanik and collaborators (2007) gave the behavior for large coupling constant of the box dimension of the spectrum of a one-dimensional discrete Schrödinger operator whose potential is a Sturm sequence associated with the golden ratio. They also show that in this case the Hausdorff and box dimensions coincide (i.e. the spectrum is dimension-regular). This Note aims at giving a simpler proof of the asymptotic property result and to generalize it to the case of any Sturm potential associated with an irrational frequency whose continued fraction expansion has bounded partial quotients. Moreover, we determine the upper box dimension of the spectrum, with large coupling constant, and show that it is not dimension-regular in general. To cite this article: Q.-H. Liu et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Dynamical upper bounds for one-dimensional quasicrystals

Following the Killip–Kiselev–Last method, we prove quantum dynamical upper bounds for discrete one-dimensional Schrödinger operators with Sturmian potentials. These bounds hold for sufficiently large coupling, almost every rotation number, and every phase.

Half-line eigenfunction estimates and purely singular continuous spectrum of zero Lebesgue measure

We consider discrete one-dimensional Schrodinger operators with minimally ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue measures is further elucidated. We provide a unified approach to both the study of the spectral type as well as the measure of the spectrum as a set. Namely, we define a stability set and show that if this set has positive measure, then it implies both absence of eigenvalues almost surely and zero-measure spectrum. As a byproduct we get absence of eigenvalues inside the original spectrum for local perturbations of these operators. We apply this approach to Schrodinger operators with Sturmian potentials. Finally, in the appendix, we discuss the two different strictly ergodic dynamical systems associated to a circle map.

Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Let β∈(0,1) be an irrational, and [a1,a2,...] be the continued fraction expansion of β. Let Hβ be the one-dimensional Schrodinger operator with Sturmian potentials. We show that if the potential strength V>20, then the Hausdorff dimension of the spectrum σ(Hβ) is strictly great than zero for any irrational β, and is strictly less than 1 if and only if lim inf k→∞(a1a2⋅⋅⋅ak))1/k<∞.

The Landauer Resistivity on Quantum Wires

We study the Landauer resistivity of the Kronig–Penney model which has various behavior depending on the potential and the Fermi energy. In the case of the Sturmian quasiperiodic potential, we discuss examples in which lim inf of it is zero.

Log-dimensional spectral properties of one-dimensional quasicrystals

We consider discrete one-dimensional Schrodinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to log-Hausdorff measures. We apply this result to operators with Sturmian potentials and thereby prove logarithmic quantum dynamical lower bounds for all coupling constants and almost all rotation numbers, uniformly in the phase.

Uniform Spectral Properties of One-Dimensional Quasicrystals, I. Absence of Eigenvalues

We consider discrete one-dimensional Schr\"odinger operators with Sturmian potentials. For a full-measure set of rotation numbers including the Fibonacci case we prove absence of eigenvalues for all elements in the hull.

Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

We study the spectral properties of discrete one-dimensional Schr\"odinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely $\alpha$-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.

Uniform Spectral Properties of One-Dimensional Quasicrystals, II. The Lyapunov Exponent

In this Letter we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schrodinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For irrational rotation numbers with bounded continued fraction expansion, it gives uniform existence of the Lyapunov exponent on the whole complex plane. Moreover, it yields uniform polynomial upper bounds on the growth rate of transfer matrices for irrational rotation numbers with bounded density. In particular, all our results apply to the Fibonacci case.