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Sum rules for Jacobi matrices and their applications to spectral theory. (English) Zbl 1050.47025

Let J be a bounded and self-adjoint Jacobi matrix with spectral measure μ 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 1 and b n 0 as n. One of the main results provides a complete characterization of the Hilbert-Schmidt perturbations

n (a n -1) 2 + n b n 2 <

in terms of the spectral measure: the absolutely continuous component μ ac of μ 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:
47B36Jacobi (tridiagonal) operators (matrices) and generalizations
47B15Hermitian and normal operators