On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices

Details On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices

2010-07-28

arxiv.org/abs/1007.5033v2

J. Approx. Theory 163 (2011), pp. 491-504

Mathematics

Spectral Theory

18pp, fixed typos (incl. one in title)

Scientific paper

We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $\Sigma_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of $\Sigma_{ac}$ which takes into account the value distribution of the diagonal elements, and implies the bound due to Deift-Simon and Poltoratski-Remling. Second, we generalise the differential inequality of Deift-Simon for the integrated density of states associated with the absolutely continuous spectrum to general Jacobi matrices.

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