David Damanik, Dirk Hundertmark, Barry Simon
Bound States and the Szeg\H{o} Condition for Jacobi Matrices and Schr\"odinger Operators
(59K, LaTeX)

ABSTRACT.  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 Szeg\H{o} condition. We provide a new proof of Nevai's result that if $\gamma >\f12$, the Szeg\H{o} condition holds, which works also if one replaces $(-1)^n$ by $\cos (\mu n)$. We show that if $\alpha =0$, $\beta\neq 0$, and $\gamma <\f12$, the Szeg\H{o} condition fails. 
We also show that if $\gamma =1$, $\alpha$ and $\beta$ are small enough 
($\beta^2 + 8 \alpha^2 < \f{1}{24}$ will do), then the Jacobi matrix has finitely many bound states (for $\alpha =0$, $\beta$ large, it has infinitely many).