  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.
