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

Abstract

For Jacobi matrices with a_n=1+(−1)^nαn−γ, b_n=(−1)^nβn−γ, we study bound states and the Szegő condition. We provide a new proof of Nevai's 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 (β^2+8α^2<1/24 will do), then the Jacobi matrix has finitely many bound states (for α=0, β large, it has infinitely many).