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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 \(\mu\) 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, \(a_n\to1\) and \(b_n\to0\) as \(n\to\infty\). One of the main results provides a complete characterization of the Hilbert-Schmidt perturbations \[ \sum_n (a_n-1)^2+\sum_n b_n^2<\infty \] 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.

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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