Sum rules for Jacobi matrices and divergent Lieb-Thirring sums

Let Ej be the eigenvalues outside [-2,2] of a Jacobi matrix with an-1∈ℓ2 and bn→0, and μ′ the density of the a.c. part of the spectral measure for the vector δ1. We show that if bn∉ℓ4, bn+1-bn∈ℓ2, then∑j(|Ej|-2)5/2=∞and if bn∈ℓ4, bn+1-bn∉ℓ2, then∫-22ln(μ′(x))(4-x2)3/2dx=-∞.We also show that if an-1,bn∈ℓ3, then the above integral is finite if and only if an+1-an,bn+1-bn∈ℓ2. We prove these and other results by deriving sum rules in which the a.c. part of the spectral measure and the eigenvalues appear on opposite sides of the equation.