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 ∈ 2 and bn → 0, and the density of the a.c. part of the spectral measure for the vector 1. We show that if bn /∈ 4, bn+1 − bn ∈ 2, then j (|Ej| − 2)5/2 =∞ and if bn ∈ 4, bn+1 − bn /∈ 2, then 2 −2 ln( (x))(4 − x2)3/2 dx =−∞. We also show that if an − 1, bn ∈ 3, then the above integral is finite if and only if an+1 − an, bn+1 − bn ∈ 2. 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. © 2005 Elsevier Inc. All rights reserved.