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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.

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