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On a spectral property of Jacobi matrices
Authors:S. Kupin
Affiliation:Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
Abstract:Let $J$ be a Jacobi matrix with elements $b_k$ on the main diagonal and elements $a_k$ on the auxiliary ones. We suppose that $J$ is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of $J$ coincides with $[-2,2]$, and its discrete spectrum is a union of two sequences ${x^pm_j}, x^+_j>2, x^-_j<-2$, tending to $pm2$. We denote sequences ${a_{k+1}-a_k}$ and ${a_{k+1}+a_{k-1}-2a_k}$ by $partial a$ and $partial^2 a$, respectively.

The main result of the note is the following theorem.

Theorem.     Let $J$ be a Jacobi matrix described above and $sigma$ be its spectral measure. Then $a-1,bin l^4, partial^2 a,partial^2 b in l^2$ if and only if

begin{displaymath}{i)} int^2_{-2} log sigma'(x) (4-x^2)^{5/2}, dx>-infty,qquad {ii)} sum_j(x^pm_jmp2)^{7/2}<infty. end{displaymath}

Keywords:Jacobi matrices   sum rules
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