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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.

47B36Jacobi (tridiagonal) operators (matrices) and generalizations
47B15Hermitian and normal operators
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