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78
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.
Dimensional Hausdorff Properties of Singular Continuous Spectra
, 1995
"... We present an extension of the GilbertPearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of onedimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze the dimensional Hausdorff prope ..."
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Cited by 28 (6 self)
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We present an extension of the GilbertPearson theory of subordinacy, which relates dimensional Hausdorff spectral properties of onedimensional Schrödinger operators to the behavior of solutions of the corresponding Schrödinger equation. We use this theory to analyze the dimensional Hausdorff properties for several examples having singularcontinuous spectrum, including sparse barrier potentials, the almost Mathieu operator and the Fibonacci Hamiltonian.
Equilibrium measures and capacities in spectral theory
, 2007
"... This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators wh ..."
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Cited by 28 (7 self)
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This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete–Szegő theory.
Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian
, 2010
"... We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prov ..."
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Cited by 23 (10 self)
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We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by EvenDar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to
Singular continuous spectrum for the Laplacian on certain sparse trees
 Commun. Math. Phys
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The canopy graph and level statistics for random operators on trees. Waves on the Regular Tree 29
 Math. Phys. Anal. Geom
, 2007
"... For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectru ..."
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Cited by 19 (1 self)
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For operators with homogeneous disorder, it is generally expected that there is a relation between the spectral characteristics of a random operator in the infinite setup and the distribution of the energy gaps in its finite volume versions, in corresponding energy ranges. Whereas pure point spectrum of the infinite operator goes along with Poisson level statistics, it is expected that purely absolutely continuous spectrum would be associated with gap distributions resembling the corresponding random matrix ensemble. We prove that on regular rooted trees, which exhibit both spectral types, the eigenstate point process has always Poissonian limit. However, we also find that this does not contradict the picture described above if that is carefully interpreted, as the relevant limit of finite trees is not the infinite tree graph but rather what is termed here the canopy graph. For this tree graph, the random Schrödinger operator is proven here to have only purepoint spectrum at any strength of the disorder.
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