Proceedings of the American Mathematical Society

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}

References [Enhancements On Off] (What's this?)

  • 1. K. M. Case, Orthogonal polynomials from the viewpoint of scattering theory, J. Mathematical Phys. 15 (1974), 2166–2174. MR 0353860
  • 2. K. M. Case, Orthogonal polynomials. II, J. Mathematical Phys. 16 (1975), 1435–1440. MR 0443644
  • 3. S. Denisov, On the Nevai's conjecture and Rakhmanov's theorem for Jacobi matrices, preprint.
  • 4. R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Annals of Math. (2) 158 (2003), 253-321.
  • 5. A. Laptev, S. Naboko, and O. Safronov, On new relations between spectral properties of Jacobi matrices and their coefficients, to appear.
  • 6. S. Molchanov, M. Novitskii, and B. Vainberg, First KdV integrals and absolutely continuous spectrum for 1-D Schrödinger operator, Comm. Math. Phys. 216 (2001), no. 1, 195–213. MR 1810778, 10.1007/s002200000333
  • 7. O. Safronov, The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation, submitted.
  • 8. B. Simon and A. Zlatos, Sum rules and the Szego condition for orthogonal polynomials on the real line, submitted.
  • 9. Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B36, 42C05

Retrieve articles in all journals with MSC (2000): 47B36, 42C05

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