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 ∈ ` 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 / ∈ `, bn+1 − bn ∈ `, then ∑ j (|Ej | − 2) = ∞, and if bn ∈ `, bn+1 − bn / ∈ `, then ∫ 2 −2 ln(μ′(x))(4− x) dx = −∞. We also show that if an − 1, bn ∈ `, then the above integral is finite if and only if an+1 − an, bn+1 − bn ∈ `. 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.