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124
General Orthogonal Polynomials
 in “Encyclopedia of Mathematics and its Applications,” 43
, 1992
"... Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed. ..."
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Cited by 91 (8 self)
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Abstract In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
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.
LiebThirring Inequalities for Jacobi Matrices
 J. APPROX. THEORY
, 2001
"... For a Jacobi matrix J on # (Z+) with Ju(n) = an1u(n 1) + bnu(n) + anu(n + 1), we prove that # E>2 n bn  n an  1. We also prove bounds on higher moments and some related results in higher dimension. ..."
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Cited by 41 (17 self)
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For a Jacobi matrix J on # (Z+) with Ju(n) = an1u(n 1) + bnu(n) + anu(n + 1), we prove that # E>2 n bn  n an  1. We also prove bounds on higher moments and some related results in higher dimension.
Sum rules and the Szegő condition for orthogonal polynomials on the real line
, 2002
"... We study the Case sum rules, especially C 0, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if lim n(an 1) = and lim nbn = exist and 2 jj, th ..."
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Cited by 34 (17 self)
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We study the Case sum rules, especially C 0, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if lim n(an 1) = and lim nbn = exist and 2 jj, then the Szegő condition fails.
CMV matrices: Five years after
, 2007
"... CMV matrices are the unitary analog of Jacobi matrices; we review their general theory. ..."
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Cited by 34 (2 self)
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CMV matrices are the unitary analog of Jacobi matrices; we review their general theory.
Jost functions and Jost solutions for Jacobi matrices, II. Decay and Analyticity
"... Abstract. We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Jost solution. We also prove L2 convergence of Szegő as ..."
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Cited by 33 (15 self)
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Abstract. We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Jost solution. We also prove L2 convergence of Szegő asymptotics on the spectrum. 1.
Diffusion and mixing in fluid flow
 Ann. of Math
"... Abstract. We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the f ..."
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Cited by 32 (7 self)
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Abstract. We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient condition on such flows is expressed naturally in terms of the spectral properties of the dynamical system associated with the flow. In particular, we find that weakly mixing flows always enhance dissipation in this sense. The proofs are based on a general criterion for the decay of the semigroup generated by an operator of the form Γ + iAL with a negative unbounded selfadjoint operator Γ, a selfadjoint operator L, and parameter A ≫ 1. In particular, they employ the RAGE theorem describing evolution of a quantum state belonging to the continuous spectral subspace of the hamiltonian (related to a classical theorem of Wiener on Fourier transforms of measures). Applications to quenching in reactiondiffusion equations are also considered. 1.
Absolutely continuous spectrum of multidimensional Schrödinger operator
 Int. Math. Res. Not
"... Abstract. We prove that 3dimensional Schrödinger operator with slowly decaying potential has an a.c. spectrum that fills R +. Asymptotics of Green’s functions is obtained as well. Consider the Schrödinger operator H = − ∆ + V, x ∈ R d We are interested in finding the support of an a.c. spectrum of ..."
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Cited by 22 (11 self)
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Abstract. We prove that 3dimensional Schrödinger operator with slowly decaying potential has an a.c. spectrum that fills R +. Asymptotics of Green’s functions is obtained as well. Consider the Schrödinger operator H = − ∆ + V, x ∈ R d We are interested in finding the support of an a.c. spectrum of H for the slowly decaying potential V. The following conjecture is due to B. Simon [21] Conjecture. If V (x) is such that R d
Halfline Schrödinger operators with no bound states
, 2003
"... We consider Schödinger operators on the halfline, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if ∆+V has no spectrum outside of the interval [−2,2], then it has purely ..."
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Cited by 21 (8 self)
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We consider Schödinger operators on the halfline, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if ∆+V has no spectrum outside of the interval [−2,2], then it has purely absolutely continuous spectrum. In the continuum case we show that if both −∆+V and −∆−V have no spectrum outside [0, ∞), then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.
Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials
 Comm. Math. Phys
"... ABSTRACT. The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,∞). Our main theorem states that this property is preserved for slowly decaying potentials provi ..."
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Cited by 18 (1 self)
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ABSTRACT. The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,∞). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables. 1.