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67
Uniform Spectral Properties Of OneDimensional Quasicrystals, IV. QuasiSturmian Potentials
 I. Absence of eigenvalues, Commun. Math. Phys
, 2000
"... We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, ..."
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Cited by 79 (44 self)
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We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ffcontinuous spectrum. All these results hold uniformly on the hull generated by a given potential.
Operators With Singular Continuous Spectrum, V. Sparse Potentials
 TO APPEAR: PROC. AMER. MATH. SOC.
, 1995
"... By presenting simple theorems for the absence of positive eigenvalues for certain onedimensional Schrödinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum. ..."
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Cited by 79 (10 self)
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By presenting simple theorems for the absence of positive eigenvalues for certain onedimensional Schrödinger operators, we are able to construct explicit potentials which yield purely singular continuous spectrum.
The absolutely continuous spectrum of Jacobi matrices
"... Abstract. I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of onedimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potential ..."
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Cited by 60 (10 self)
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Abstract. I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of onedimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a DenisovRakhmanov type theorem for the general finite gap case. The main theme is the following: It is extremely difficult to produce absolutely continuous spectrum in one space dimension and thus its existence has strong implications.
αContinuity Properties of OneDimensional Quasicrystals
 COMMUN. MATH. PHYS
, 1997
"... We apply the JitomirskayaLast extension of the GilbertPearson theory to discrete onedimensional Schrödinger operators with potentials arising from generalized Fibonacci sequences. We prove for certain rotation numbers that for every value of the coup ling constant, there exists an α > 0 such t ..."
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Cited by 31 (22 self)
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We apply the JitomirskayaLast extension of the GilbertPearson theory to discrete onedimensional Schrödinger operators with potentials arising from generalized Fibonacci sequences. We prove for certain rotation numbers that for every value of the coup ling constant, there exists an α > 0 such that the corresponding operator has purely ffcontinuous spectrum. This result follows from uniform upper and lower bounds for the k \Delta kLnorm of the solutions corresponding to energies from the spectrum of the operator.
Strictly ergodic subshifts and associated operators
, 2005
"... We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have in ..."
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Cited by 30 (17 self)
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We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.
The fractal dimension of the spectrum of the Fibonacci Hamiltonian
 COMMUN. MATH
, 2008
"... 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(Hλ))·log λ converges to an explicit constant ( ≈ 0.88137). We also discuss consequences of these results for the rate o ..."
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Cited by 30 (15 self)
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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(Hλ))·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.
PowerLaw Bounds On Transfer Matrices And Quantum Dynamics In One Dimension
"... We present an approach to quantum dynamical lower bounds for discrete onedimensional Schrodinger operators which is based on powerlaw bounds on transfer matrices. It suces to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamil ..."
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Cited by 25 (17 self)
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We present an approach to quantum dynamical lower bounds for discrete onedimensional Schrodinger operators which is based on powerlaw bounds on transfer matrices. It suces to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.
Singular spectrum of Lebesgue measure zero for quasicrystals
 Commun. Math. Phys
, 2002
"... exponent, linear repetitivity, primitive substitution Abstract. The spectrum of onedimensional 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. T ..."
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Cited by 21 (5 self)
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exponent, linear repetitivity, primitive substitution Abstract. The spectrum of onedimensional 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 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 as e.g. the RudinShapiro 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. 1.
Palindrome complexity
 To appear, Theoret. Comput. Sci
, 2002
"... We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points o ..."
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Cited by 19 (2 self)
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We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence. We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to “class P ” of HofKnillSimon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block)complexity. 1