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

Damanik, David; Lenz, Daniel

doi:10.1007/s002200050742

Cited by 104 publications

(261 citation statements)

References 26 publications

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“…The section on almost periodic operators summarizes some results from [21] and [122]. There have recently been some efforts to prove uniform spectral results (i.e, for all operators in the hull), see, for example, [51], [52] and the references therein. In this respect, it should be mentioned that the absolutely continuous spectrum is constant for all operators in the hull ( [163], Theorem 1.5).…”

Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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Section: Notes On Literaturementioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

323

349

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“…1 It was later seen that one also has the absence of point spectrum for all parameter values; see Sütő [34], Hof-Knill-Simon [17], and Kaminaga [21] for partial results and Damanik-Lenz [9] for the full result. Thus, the Fibonacci model exhibits purely singular continuous spectrum that is very rigid in the sense that it is not affected by a change of the defining parameters.…”

Section: Introductionmentioning

confidence: 99%

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

Damanik

Embree

Gorodetski

et al. 2008

Commun. Math. Phys.

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Abstract. We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as λ → ∞, dim(σ(H λ )) · log λ converges to an explicit constant (≈ 0.88137). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.

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“…We mention that quasiperiodic sequences serve as models for one-dimensional quasicrystals and their sometimes exotic transport properties. Especially, the discrete one-body Schrödinger operator with Fibonacci potential, see (5), has been considered [10,[12][13][14][15][16][17][18][19][20][21][22][23][24]. Quasiperiodic spin chains (in particular with Fibonacci disorder) have also been studied extensively, with a focus on spectral properties and critical phenomena [25][26][27][28][29][30][31][32].…”

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confidence: 99%

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

et al. 2014

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We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone jxj ≤ vjtj, we obtain jxj ≤ vjtj α for some 0 < α < 1. We can characterize the allowed values of α exactly as those exceeding the upper transport exponent α þ u of a one-body Schrödinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field. Introduction.-Relativistic systems are local in the sense that information propagates at most at the speed of light. In their seminal paper [1], Lieb and Robinson found that nonrelativistic quantum spin systems described by local Hamiltonians satisfy a similar "quasilocality" under the Heisenberg dynamics. Their Lieb-Robinson (LR) bound and its recent generalizations [2,3] implies the existence of a "light cone" jxj ≤ vjtj in space-time, outside of which quantum correlations (concretely: commutators of local observables) are exponentially small. In other words, the LR bound shows that, to a good approximation, quantum correlations propagate at most ballistically, with a systemdependent "Lieb-Robinson velocity" v.About ten years ago, the general interest in LR bounds resurged when Hastings and coworkers realized that they are the key tool for deriving exponential clustering, a higher-dimensional Lieb-Schultz-Mattis theorem, and the celebrated area law for the entanglement entropy in onedimensional systems with a spectral gap [2,4,5]. These results highlight the role of entanglement in constraining the structure of ground states in gapped systems and yield many applications to quantum information theory, e.g., in developing algorithms to simulate quantum systems on a classical computer [6,7].In this Letter, we announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. The LR bound is anomalous in the sense that the forward half of the ordinary light cone is changed to the region jxj ≤ vjtj α for some 0 < α < 1.Previous study has focused on the dependence of the Lieb-Robinson velocity v on the system details [3], with particular interest in the case v ¼ 0, since it may be interpreted as dynamical localization [8]. In a very recent paper [9], a logarithmic light cone was obtained for long-range, i.e., power-law decaying, interactions. The

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

Abstract: We consider discrete one-dimensional Schrödinger operators with Sturmian potentials. For a fullmeasure set of rotation numbers including the Fibonacci case we prove absence of eigenvalues for all elements in the hull.

Cited by 104 publications

References 26 publications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Jacobi Operators and Completely Integrable Nonlinear Lattices

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

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