Bound States and the Szego Condition for Jacobi Matrices and Schrodinger Operators

Details Bound States and the Szego Condition for Jacobi Matrices and Schrodinger Operators Bound States and the Szego Condition for Jacobi Matrices and Schrodinger Operators

2002-08-25

arxiv.org/abs/math-ph/0208035v1

Physics

Mathematical Physics

22 pages

Scientific paper

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).

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