Bound States and the Szego Condition for Jacobi Matrices and Schrodinger   Operators
  For Jacobi matrices with a_n = 1+(-1)^n alpha n^{-gamma}, b_n = (-1)^n beta n^{-gamma}, we study bound states and the SzegHo condition. We provide a new proof of Nevai's result that if gamma > 1/2, the Szego condition holds, which works also if one replaces (-1)^n by cos(mu n). We show that if alpha = 0, beta not equal to 0, and gamma < 1/2, the Szego condition fails. We also show that if gamma = 1, alpha and beta are small enough (beta^2 + 8 alpha^2 < 1/24 will do), then the Jacobi matrix has finitely many bound states (for alpha = 0, beta large, it has infinitely many). 
David Damanik (Caltech), Dirk Hundertmark (Caltech), Barry Simon   (Caltech)