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59
The absolutely continuous spectrum of onedimensional Schrödinger operators with decaying potentials
, 2008
"... This paper deals with general structural properties of onedimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectr ..."
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Cited by 63 (9 self)
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This paper deals with general structural properties of onedimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and DenisovRakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.
Perturbations of orthogonal polynomials with periodic recursion coefficients
, 2007
"... We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well ada ..."
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Cited by 45 (16 self)
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We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.
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.
REFLECTIONLESS HERGLOTZ FUNCTIONS AND GENERALIZED LYAPUNOV EXPONENTS
"... Abstract. We study several related aspects of reflectionless Jacobi matrices. Our first set of results deals with the singular part of reflectionless measures. We then introduce and discuss Lyapunov exponents, density of states measures, and other related quantities in a general setting. This is rel ..."
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Cited by 24 (8 self)
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Abstract. We study several related aspects of reflectionless Jacobi matrices. Our first set of results deals with the singular part of reflectionless measures. We then introduce and discuss Lyapunov exponents, density of states measures, and other related quantities in a general setting. This is related to the previous material because the density of states measures are reflectionless on certain sets. 1.
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
The ChristoffelDarboux Kernel
"... A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results. ..."
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Cited by 19 (5 self)
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A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.
Finite gap Jacobi matrices, I. The isospectral torus
, 2008
"... Let e ⊂ R be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent ..."
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Cited by 16 (14 self)
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Let e ⊂ R be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.
Absolutely continuous spectrum for the almost Mathieu operator
, 2008
"... We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century. ..."
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Cited by 15 (5 self)
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We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century.
The Nevai Condition
, 2010
"... We study Nevai’s condition that for orthogonal polynomials on the real line, Kn(x, x0) 2 Kn(x0,x0) −1 dρ(x) → δx0, where Kn is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on ..."
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Cited by 13 (4 self)
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We study Nevai’s condition that for orthogonal polynomials on the real line, Kn(x, x0) 2 Kn(x0,x0) −1 dρ(x) → δx0, where Kn is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2, 2] where it fails on an interval.
Spectral properties of reflectionless Jacobi CMV, and Schrödinger operators

, 2008
"... We prove that Jacobi, CMV, and Schrödinger operators, which are reflectionless on a homogeneous set E (in the sense of Carleson), under the assumption of a Blaschketype condition on their discrete spectra accumulating at E, have purely absolutely continuous spectrum on E. ..."
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Cited by 10 (3 self)
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We prove that Jacobi, CMV, and Schrödinger operators, which are reflectionless on a homogeneous set E (in the sense of Carleson), under the assumption of a Blaschketype condition on their discrete spectra accumulating at E, have purely absolutely continuous spectrum on E.