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On a spectral property of Jacobi matrices

Author: S. Kupin
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1377-1383
MSC (2000): Primary 47B36; Secondary 42C05
Published electronically: December 12, 2003
MathSciNet review: 2053342
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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 $a-1,b\in l^4, \partial^2 a,\partial^2 b \in l^2$ 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_j\mp2)^{7/2}<\infty. \end{displaymath}

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Additional Information

S. Kupin
Affiliation: Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912

Keywords: Jacobi matrices, sum rules
Received by editor(s): October 25, 2002
Published electronically: December 12, 2003
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society