*(English)*Zbl 1050.47025

Let $J$ be a bounded and self-adjoint Jacobi matrix with spectral measure $\mu $ and entries ${b}_{n}$ along the main diagonal and ${a}_{n}$ along two others. The authors undertake a thorough investigation of those $J$’s which are compact perturbations of the free matrix (discrete Laplacian) ${J}_{0}$, that is, ${a}_{n}\to 1$ and ${b}_{n}\to 0$ as $n\to \infty $. One of the main results provides a complete characterization of the Hilbert-Schmidt perturbations

in terms of the spectral measure: the absolutely continuous component ${\mu}_{ac}$ of $\mu $ obeys the quasi-Szegő condition and the eigenvalues off the essential spectrum $[-2,2]$ tend to the endpoints with a certain rate. The authors also prove Nevai’s conjecture which claims that the Szegő condition holds as long as $J$ is a trace class perturbation of ${J}_{0}$. The key to the proofs is a family of equalities called the Case sum rules, with the terms on the left-hand side purely spectral and those on the right depending in a simple way on the matrix entries. Of particular interest is a certain combination of the sum rules with the property that each of its terms is nonnegative.

##### MSC:

47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |

47B15 | Hermitian and normal operators |