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78
Operators with singular continuous spectrum, II: Rank one operators
 J. ANAL. MATH
, 1996
"... For an operator, A, with cyclic vector ϕ, we study A + λP where P is the rank one projection onto multiples of ϕ. If [α, β] ⊂ spec(A) andA has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. ..."
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Cited by 179 (32 self)
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For an operator, A, with cyclic vector ϕ, we study A + λP where P is the rank one projection onto multiples of ϕ. If [α, β] ⊂ spec(A) andA has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s.
Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators
, 1999
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Quantum Dynamics and Decompositions of Singular Continuous Spectra
 J. Funct. Anal
, 1995
"... . We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (wi ..."
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Cited by 104 (10 self)
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. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on "fractal" properties) and the nature of the time evolution have been the subject of several recent papers [7,13,1518,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...
Modified Prüfer and EFGP Transforms and the Spectral Analysis of OneDimensional Schrödinger Operators
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1997
"... Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete halfline Schrödinger operators with slowly decaying potentials. Among our results we show if V (x) = ∑∞ n=1 anW (x − xn), where W has compact support and xn/xn+1 → 0, then H has purely ..."
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Cited by 86 (24 self)
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Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete halfline Schrödinger operators with slowly decaying potentials. Among our results we show if V (x) = ∑∞ n=1 anW (x − xn), where W has compact support and xn/xn+1 → 0, then H has purely a.c. (resp. purely s.c.) spectrum on (0, ∞) if ∑ a2 n <∞(resp. ∑ a2 n = ∞). For λn−1/2an potentials, where an are independent, identically distributed random variables with E(an) =0,E(a2 n)=1,and λ < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.
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.
Singular Continuous Spectrum for Palindromic Schrödinger Operators
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1995
"... We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies th ..."
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Cited by 66 (5 self)
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We give new examples of discrete Schrödinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such an x is that there is a z G X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in X. The class includes wellknown substitutions like Fibonacci, ThueMorse, Period Doubling, binary nonPisot and ternary nonPisot. We also show that the operator has no absolutely continuous spectrum for all x G X if X derives from a primitive substitution. For potentials defined by circle maps, x n = ly($o + nu), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational α and every halfopen interval J.
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.
Perturbations of onedimensional Schrödinger operators preserving the absolutely continuous spectrum
, 2002
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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.