Bound states and the Szegő condition for Jacobi matrices and Schrödinger operators

For Jacobi matrices with an=1+(−1)nαn−γ, bn=(−1)nβn−γ, we study bound states and the Szegő condition. We provide a new proof of Nevaiʹs result that if γ>12, the Szegő condition holds, which works also if one replaces (−1)n by cos (μn). We show that if α=0, β≠0, and γ<12, the Szegő condition fails. We also show that if γ=1, α and β are small enough (β2+8α2<124 will do), then the Jacobi matrix has finitely many bound states (for α=0, β large, it has infinitely many).