- 03-530 David Damanik, Rowan Killip, Barry Simon
- Necessary and Sufficient Conditions in the Spectral Theory of
Jacobi Matrices and Schr"odinger Operators
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Dec 9, 03
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Abstract. We announce three results in the theory of Jacobi matrices
and Schr\"odinger operators. First, we give necessary and
sufficient conditions for a measure to be the spectral measure of
a Schr\"odinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2
(0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary
condition. Second, we give necessary and sufficient conditions on
the Jacobi parameters for the associated orthogonal polynomials to
have Szeg\H{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.
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