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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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