**
Below is the ascii version of the abstract for 03-530.
The html version should be ready soon.**David Damanik, Rowan Killip, Barry Simon
Necessary and Sufficient Conditions in the Spectral Theory of
Jacobi Matrices and Schr"odinger Operators
(30K, LaTeX)
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.