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.