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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 105 (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...
An invitation to random Schrödinger operators
, 2007
"... This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are m ..."
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Cited by 83 (8 self)
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This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.
Metalinsulator transition for the almost Mathieu operator
, 1999
"... We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+1)+Ψ(n −1)+λcos2π(ωn+θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the AubryAndré conjecture. ..."
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Cited by 67 (8 self)
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We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+1)+Ψ(n −1)+λcos2π(ωn+θ)Ψ(n), exhibits localization for λ> 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the AubryAndré conjecture.
Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles
 Arn61] [AS83] V. I. Arnol ′ d. Small denominators. I. Mapping the
, 2003
"... Abstract. We show that for almost every frequency α ∈ R \ Q, for every C ω potential v: R/Z → R, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic (similar results are valid in the smooth category). We describe several a ..."
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Cited by 58 (11 self)
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Abstract. We show that for almost every frequency α ∈ R \ Q, for every C ω potential v: R/Z → R, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic (similar results are valid in the smooth category). We describe several applications for the quasiperiodic Schrödinger operator, including persistence of absolutely continuous spectrum under perturbations of the potential. Such results also allow us to complete the proof of the AubryAndré conjecture on the measure of the spectrum of the Almost Mathieu Operator.
The Ten Martini Problem
"... Abstract. We prove the conjecture (known as the \Ten Martini Problem " after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies. 1. ..."
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Cited by 51 (7 self)
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Abstract. We prove the conjecture (known as the \Ten Martini Problem " after Kac and Simon) that the spectrum of the almost Mathieu operator is a Cantor set for all nonzero values of the coupling and all irrational frequencies. 1.
Duality and singular continuous spectrum in the almost Mathieu equation
, 1997
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Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential.
, 2001
"... this paper we study continuity of the Lyapunov exponent associated with 1D quasiperiodic operators. Assume v real analytic on T. Let v : T ! R : Consider an SL 2 (R) valued function A(x; E) = v(x) E 1 1 0 ; x 2 T: (1.1) Set MN (E; x; !) = A(S j x); Sx = x + !; LN (E; !) = log kMN ..."
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Cited by 45 (8 self)
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this paper we study continuity of the Lyapunov exponent associated with 1D quasiperiodic operators. Assume v real analytic on T. Let v : T ! R : Consider an SL 2 (R) valued function A(x; E) = v(x) E 1 1 0 ; x 2 T: (1.1) Set MN (E; x; !) = A(S j x); Sx = x + !; LN (E; !) = log kMN (E; x; !)kdx: The Lyapunov exponent is de ned by L(E; !) = lim LN (E) = inf LN (E; !) and exists by subadditivity
The integrated density of states for random Schrödinger operators
"... We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed i ..."
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Cited by 43 (4 self)
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We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems