摘要 :
We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi mat...
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We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators?
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摘要 :
We study structural properties of the Lyapunov exponent γ and the density of states k for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function w = − γ + ...
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We study structural properties of the Lyapunov exponent γ and the density of states k for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function w = − γ + iπ k as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.
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We discuss the spectral properties of higher order ordinary differential operators. If the coefficients differ from constants by small perturbations, then the spectral properties are preserved. In this context, "small perturbation...
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We discuss the spectral properties of higher order ordinary differential operators. If the coefficients differ from constants by small perturbations, then the spectral properties are preserved. In this context, "small perturbations" are either short range (i.e., integrable) or long range, but slowly varying. This generalizes classical results on second order operators. Our approach relies on an analysis of the associated differential equations with the help of uniform asymptotic integration techniques.
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摘要 :Consider the Schrödinger operator H=−d 2/dx 2+V(x) with power-decaying potential V(x)=O(x −α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity...
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Consider the Schrödinger operator H=−d 2/dx 2+V(x) with power-decaying potential V(x)=O(x −α). We prove that a previously obtained dimensional bound on exceptional sets of the WKB method is sharp in its whole range of validity. The construction relies on pointwise bounds on finite gap potentials. These bounds are obtained by an analysis of the Jacobi inversion problem on hyperelliptic Riemann surfaces.
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We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional Schrödinger operators with sparse potentials. We find a resonance structure which admits a physical interpretation in t...
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We study the pointwise behavior of the Fourier transform of the spectral measure for discrete one-dimensional Schrödinger operators with sparse potentials. We find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model. We also present an improved version of known results on the spectrum of such operators.
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We link recently developed approaches to the inverse spectral problem (due to Simon and myself, respectively). We obtain a description of the set of Simon’s A functions in terms of a positivity condition. This condition also char...
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We link recently developed approaches to the inverse spectral problem (due to Simon and myself, respectively). We obtain a description of the set of Simon’s A functions in terms of a positivity condition. This condition also characterizes the solubility of Simon’s fundamental equation.
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We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation (p_0y") + (p_1y')' + qy = zwy subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asympto...
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We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation (p_0y") + (p_1y')' + qy = zwy subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asymptotic integration and the Titchmarsh-Weyl M-matrix. Asymptotic integration yields asymptotic formulae for the solutions of the differential equation which are then used to derive properties of the M-matrix. The characterisation of spectral properties in terms of the boundary behaviour of M leads to the desired results.
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摘要 :
We investigate one-dimensional Schrodinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embe...
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We investigate one-dimensional Schrodinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.
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摘要 :
We study several related aspects of reflectionless Jacobi matrices. First, we discuss the singular part of the corresponding spectral measures. We then show how to identify sets on which measures are reflectionless by looking at t...
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We study several related aspects of reflectionless Jacobi matrices. First, we discuss the singular part of the corresponding spectral measures. We then show how to identify sets on which measures are reflectionless by looking at the logarithmic potentials of these measures. Communicated by B. Simon
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