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19
A characterization of the Anderson metalinsulator transport transition
 Duke Math. J
"... We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong... ..."
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Cited by 56 (19 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
Dynamical Upper Bounds On Wavepacket Spreading
 Am. J. Math
, 2001
"... We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by prope ..."
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Cited by 31 (2 self)
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We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian  the most studied onedimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.
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.
Upper bounds in quantum dynamics
 J. Amer. Math. Soc
"... Abstract. We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on timeaveraged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is a ..."
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Cited by 20 (17 self)
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Abstract. We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on timeaveraged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport. For quasiperiodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a wellknown result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two. 1.
Generalized Fractal Dimensions: Equivalences and Basic Properties
, 2000
"... Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q & ..."
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Cited by 19 (8 self)
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Given a positive probability Borel measure , we establish some basic properties of the associated functions (q) and of the generalized fractal dimensions D for q 2 R. We rst give the connections between the generalized fractal dimensions, the Renyi dimensions and the meanq dimensions when q > 0. We then use these relations to prove some regularity properties for (q); we also provide some estimates for these functions (in particular estimates on their behaviour at 1), as well as for the dimensions corresponding to convolution of two measures. We nally present some calculations for speci c examples. 1
FourierBessel Functions of Singular Continuous Measures and their Many Asymptotics
 ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
, 2006
"... We study the Fourier transform of polynomials in an orthogonal family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier–Bessel functions, in the argument, the order, and in certain combinations of the two is required to ..."
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Cited by 11 (9 self)
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We study the Fourier transform of polynomials in an orthogonal family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier–Bessel functions, in the argument, the order, and in certain combinations of the two is required to solve a number of problems arising in quantum mechanics. We present known results, new approaches and open conjectures, hoping to justify our belief that the importance of these investigations extends beyond the application just mentioned, and may involve interesting discoveries.
Mixed Lower Bounds for Quantum Transport
 J. Funct. Anal
"... this paper). Mathematicians have a dierent de nition. They say that the measure is of exact Hausdor dimension if dim () = dim () = and in the same way for packing dimensions. It is clear that not all measures have exact Hausdor or packing dimensions ..."
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Cited by 7 (6 self)
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this paper). Mathematicians have a dierent de nition. They say that the measure is of exact Hausdor dimension if dim () = dim () = and in the same way for packing dimensions. It is clear that not all measures have exact Hausdor or packing dimensions