Results 1  10
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14
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.
Anderson Localization for the Almost Mathieu Equation: A Nonperturbative Proof.
"... We prove that for any diophantine rotation angle ! and a.e. phase ` the almost Mathieu operator (H(`)\Psi) n = \Psi n\Gamma1 + \Psi n+1 + cos(2ß(` + n!))\Psi n has pure point spectrum with exponentially decaying eigenfunctions for 15: We also prove the existence of some pure point spectrum for an ..."
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Cited by 22 (9 self)
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We prove that for any diophantine rotation angle ! and a.e. phase ` the almost Mathieu operator (H(`)\Psi) n = \Psi n\Gamma1 + \Psi n+1 + cos(2ß(` + n!))\Psi n has pure point spectrum with exponentially decaying eigenfunctions for 15: We also prove the existence of some pure point spectrum for any 5:4: Permanent address: International Institute of Earthquake Prediction Theory and Mathematical Geophysics. Moscow, Russia . 1. INTRODUCTION In this paper we study localization for the almostMathieu operator on ` 2 (Z) : (H(`)\Psi) n = \Psi n\Gamma1 + \Psi n+1 + cos(2ß(` + n!))\Psi n The almostMathieu operator attracted a lot of interest especially in the last decade. For references before 1985 see [1]. Some of the later references are [27 ]. While it is very well understood and commonly believed that for diophantine ! and jj ? 2 the operator H(`) should have pure point spectrum with exponentially decaying eigenfunctions for almost every ` this is not yet rigorously proved ....
Anomalous transport: A mathematical framework
 MR 99b:81046 162
, 1998
"... We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize ..."
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Cited by 18 (8 self)
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We develop a mathematical framework allowing to study anomalous transport in homogeneous solids. The main tools characterizing the anomalous transport properties are spectral and diffusion exponents associated to the covariant Hamiltonians describing these media. The diffusion exponents characterize the spectral measures entering in Kubo’s formula for the conductivity and hence lead to anomalies in Drude’s formula. We give several formulas allowing to calculate these exponents and treat, as an example, Wegner’s norbital model as well as the Anderson model in coherent potential approximation. 1
ABSOLUTELY CONTINUOUS AND SINGULAR SPECTRAL SHIFT FUNCTIONS
, 2008
"... Given a selfadjoint operator H0, a selfadjoint trace class operator V and a fixed HilbertSchmidt operator F with trivial kernel and cokernel, using limiting absorption principle an explicit set of full Lebesgue measure Λ(H0, F) ⊂ R is defined, such that for all points of this set the wave and t ..."
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Cited by 6 (5 self)
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Given a selfadjoint operator H0, a selfadjoint trace class operator V and a fixed HilbertSchmidt operator F with trivial kernel and cokernel, using limiting absorption principle an explicit set of full Lebesgue measure Λ(H0, F) ⊂ R is defined, such that for all points of this set the wave and the scattering matrices can be defined unambiguously. Many wellknown properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This new abstract scattering theory allows to prove that the singular part of the spectral shift function is an almost everywhere integervalued function for trace class perturbations of a selfadjoint operator.
Exotic Spectra: A Review of Barry Simon’s Central Contributions
"... We review some of Barry Simon’s central contributions concerning what is often called exotic spectral properties. These include phenomena such as Cantor spectrum, thick point spectrum and singular continuous spectrum. ..."
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Cited by 5 (0 self)
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We review some of Barry Simon’s central contributions concerning what is often called exotic spectral properties. These include phenomena such as Cantor spectrum, thick point spectrum and singular continuous spectrum.
Absence of absolutely continuous spectra for multidimensional Schrodinger operators with high barriers
 Bull. London Math. Soc
, 1995
"... We prove absence of absolutely continuous spectra for multidimensional Schrodinger operators with high barriers. The result is formulated in terms of a geometric condition on the barriers which entails singular spectrum. The proof combines probabilistic and functional analytic techniques. 1. ..."
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Cited by 4 (3 self)
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We prove absence of absolutely continuous spectra for multidimensional Schrodinger operators with high barriers. The result is formulated in terms of a geometric condition on the barriers which entails singular spectrum. The proof combines probabilistic and functional analytic techniques. 1.
The Lyapunov exponents for Schrodinger operators with slowly oscillating potentials
 J. Funct. Anal
, 1996
"... Abstract. By studying the integrated density of states, we prove the existence of Lyapunov exponents and the Thouless formula for the Schrödinger operator −d 2 /dx 2 +cosx ν with 0 <ν<1onL 2 [0, ∞). This yields an explicit formula for these Lyapunov exponents. By applying rank one perturbation ..."
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Cited by 4 (1 self)
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Abstract. By studying the integrated density of states, we prove the existence of Lyapunov exponents and the Thouless formula for the Schrödinger operator −d 2 /dx 2 +cosx ν with 0 <ν<1onL 2 [0, ∞). This yields an explicit formula for these Lyapunov exponents. By applying rank one perturbation theory, we also obtain some spectral consequences. 1.
Genericity of certain classes of unitary and selfadjoint operators, Canad
 Math. Bull
, 1998
"... ABSTRACT. In a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, form a ..."
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Cited by 4 (0 self)
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ABSTRACT. In a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, form a dense Gé. A similar theorem for bounded selfadjoint operators with a given norm bound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon’s result, with simplicity of the spectrum added. Theorem 1 (p. 148) of ChoksiNadkarni [1] (hereafter referred to as CN) states THEOREM 1. Pc, the set of nonatomic (i.e. continuous) probability measures on the circle S1, is a dense Gé in P the set of all probability measures on the circle (in the weak* topology.). Since the set of measures in P which vanish on a fixed open arc I of the circle is closed, nowhere dense in P we immediately have COROLLARY (CN P. 148). The set fñ: ñ2Pc, support(ñ) ≥ S 1 g is dense G é in P. If J is any fixed closed arc in S 1,andP(J), resp. Pc(J), denote the probability measures, resp. nonatomic probability measures, with support in J, exactly the same proof shows. THEOREM 10. Pc(J) is a dense Gé in P (J). Thesetfñ: ñ2Pc(J), support(ñ) ≥ Jg is dense Gé in P (J). In any case these results are wellknown. Note that the intersection of two dense G é sets is a dense G é since P is a Baire space. We now use Theorem 2 of CN (p. 148), which states
˜Q − REPRESENTATION OF REAL NUMBERS AND FRACTAL PROBABILITY DISTRIBUTIONS
, 2007
"... Abstract. A e Q−representation of real numbers is introduced as a generalization of the s−adic expansion. It is shown that the e Q−representation is a convenient tool for the construction and study of fractals and measures with complicated local structure. Distributions of random variables ξ with in ..."
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Cited by 1 (1 self)
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Abstract. A e Q−representation of real numbers is introduced as a generalization of the s−adic expansion. It is shown that the e Q−representation is a convenient tool for the construction and study of fractals and measures with complicated local structure. Distributions of random variables ξ with independent e Q−symbols are studied in details. Necessary and sufficient conditions for the corresponding probability measures µξ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the eQ−representation. The metric, topological, and fractal properties for the distribution of ξ are investigated. A number of examples are presented. 1 Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D