Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schr\"odinger Operators

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 $-\text{f}{d}^{2}d{x}^2+V(x)$ on $L^2(0,\mathrm{\infty })$ with $V\in L^2(0,\mathrm{\infty })$ 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.