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29 March 2024
 
  » arxiv » math.SP/0309206

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Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators
David Damanik ; Rowan Killip ; Barry Simon ;
Date 12 Sep 2003
Subject Spectral Theory; Mathematical Physics | math.SP math-ph math.MP
AbstractWe announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator $-f{d^2}{dx^2} +V(x)$ on $L^2 (0,infty)$ with $Vin L^2 (0,infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have SzegH{o} asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.
Source arXiv, math.SP/0309206
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