Sum Rules for Jacobi Matrices and Divergent Lieb-Thirring Sums

Extending earlier work of Killip-Simon and Simon-Zlatos, we obtain sum rules for Jacobi matrices in which the a.c. part of the spectral measure and the eigenvalues of the matrix appear on opposite sides of the equation. We use these to obtain various results on divergence of certain Lieb-Thirring sums of eigenvalues and Szego-type integrals of the a.c. part of the spectral measure.