Jost Functions and Jost Solutions for Jacobi Matrices, I. A Necessary   and Sufficient Condition for Szego Asymptotics
  We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szeg\H{o} asymptotics off the real axis. A key idea is to prove the equivalence of Szeg\H{o} asymptotics and of Jost asymptotics for the Jost solution. We also prove $L^2$ convergence of Szeg\H{o} asymptotics on the spectrum. 
David Damanik (Caltech), Barry Simon (Caltech)