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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a spectral property of Jacobi matrices
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by S. Kupin
Proc. Amer. Math. Soc. 132 (2004), 1377-1383
DOI: https://doi.org/10.1090/S0002-9939-03-07244-7
Published electronically: December 12, 2003

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 $\pm 2$. 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 \[ \mathrm {i)}\ \int ^2_{-2} \log \sigma ’(x) (4-x^2)^{5/2} dx>-\infty ,\qquad \mathrm {ii)}\ \sum _j(x^\pm _j\mp 2)^{7/2}<\infty . \]

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Bibliographic Information
  • S. Kupin
  • Affiliation: Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
  • Email: kupin@math.brown.edu
  • Received by editor(s): October 25, 2002
  • Published electronically: December 12, 2003
  • Communicated by: Andreas Seeger
  • © Copyright 2003 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1377-1383
  • MSC (2000): Primary 47B36; Secondary 42C05
  • DOI: https://doi.org/10.1090/S0002-9939-03-07244-7
  • MathSciNet review: 2053342