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46
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...
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.
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
Zero Measure Spectrum for the Almost Mathieu Operator
 Commun. Math. Phys
, 1993
"... . We study the almost Mathieu operator: (H ff;;` u)(n) = u(n + 1) + u(n \Gamma 1) + cos(2ßffn + `)u(n) , on l 2 (Z) , and show that for all ; ` , and (Lebesgue) a.e. ff , the Lebesgue measure of its spectrum is precisely j4 \Gamma 2jjj . In particular, for jj = 2 the spectrum is a zero measure c ..."
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Cited by 33 (3 self)
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. We study the almost Mathieu operator: (H ff;;` u)(n) = u(n + 1) + u(n \Gamma 1) + cos(2ßffn + `)u(n) , on l 2 (Z) , and show that for all ; ` , and (Lebesgue) a.e. ff , the Lebesgue measure of its spectrum is precisely j4 \Gamma 2jjj . In particular, for jj = 2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational ff 's (and jj = 2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1=2 . * Work partially supported by the GIF. 1. Introduction In this paper, we study the almost Mathieu (also called Harper's) operator on l 2 (Z) . This is the (bounded, self adjoint) operator H ff;;` , defined by: H ff;;` = H 0 + V ff;;` ; (H 0 u)(n) = u(n + 1) + u(n \Gamma 1) ; (V ff;;` u)(n) = cos(2ßffn + `)u(n) ; (1:1) where ff; ; ` 2 R . H ff;;` is a tight binding model for the Hamiltonian of an electron in a one dimensional lattice, subject to a commensurate (if ff is rational) or incommensurate (if ff is irrational) potential....
On the Measure of the Spectrum for the Almost Mathieu Operator
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1990
"... We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, α, the measure of the spectrum, S(α,Λ,,θ) = 4 — 2\λ\\. For \λ\ή=2 we show that if an is rational and αM^α irrational, then S+(απ,λ,0)»42λ. ..."
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Cited by 31 (3 self)
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We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, α, the measure of the spectrum, S(α,Λ,,θ) = 4 — 2\λ\\. For \λ\ή=2 we show that if an is rational and αM^α irrational, then S+(απ,λ,0)»42λ.
Almost everything about the almost Mathieu operator. I
 In XIth International Congress of Mathematical Physics
, 1994
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Cantor spectrum for the almost Mathieu operator
 Commun. Math. Phys
"... In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx) n = xn+1 + xn−1 + bcos (2πnω + φ) xn on l 2 (Z) and its associated eigenvalue equation to deduce that for b = 0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of th ..."
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Cited by 26 (2 self)
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In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx) n = xn+1 + xn−1 + bcos (2πnω + φ) xn on l 2 (Z) and its associated eigenvalue equation to deduce that for b = 0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the socalled “Ten Martini Problem ” for these values of b and ω. Moreover, we prove that for b  = 0 small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open. 1 Introduction. Main
Quantum Hall Effect on the hyperbolic plane
 Commun. Math. Physics
, 1997
"... Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potentia ..."
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Cited by 24 (14 self)
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Abstract. We study both the continuous model and the discrete model of the quantum Hall effect (QHE) on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian.
The history of qcalculus and a new method
, 2000
"... 1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8 ..."
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Cited by 23 (10 self)
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1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8