Let be a bounded and self-adjoint Jacobi matrix with spectral measure and entries along the main diagonal and along two others. The authors undertake a thorough investigation of those ’s which are compact perturbations of the free matrix (discrete Laplacian) , that is, and as . One of the main results provides a complete characterization of the Hilbert-Schmidt perturbations
in terms of the spectral measure: the absolutely continuous component of obeys the quasi-Szegő condition and the eigenvalues off the essential spectrum 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 is a trace class perturbation of . 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.