02-356 David Damanik, Dirk Hundertmark, Barry Simon

Bound States and the Szeg\H{o} Condition for Jacobi Matrices and Schr\"odinger Operators (59K, LaTeX) Aug 25, 02

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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).

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