On a spectral property of Jacobi matrices
Author:
S. Kupin
Translated by:
Journal:
Proc. Amer. Math. Soc. 132 (2004), 13771383
MSC (2000):
Primary 47B36; Secondary 42C05
Published electronically:
December 12, 2003
MathSciNet review:
2053342
Fulltext PDF Free Access
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Abstract: Let be a Jacobi matrix with elements on the main diagonal and elements on the auxiliary ones. We suppose that is a compact perturbation of the free Jacobi matrix. In this case the essential spectrum of coincides with , and its discrete spectrum is a union of two sequences , tending to . We denote sequences and by and , respectively. The main result of the note is the following theorem. Theorem. Let be a Jacobi matrix described above and be its spectral measure. Then if and only if
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 K. Case, Orthogonal polynomials from the viewpoint of scattering theory, J. Mathematical Phys. 15 (1974), 21662174. MR 50:6342
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 K. Case, Orthogonal polynomials, II, J. Mathematical Phys. 16 (1975), 14351440. MR 56:2008
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 S. Denisov, On the Nevai's conjecture and Rakhmanov's theorem for Jacobi matrices, preprint.
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 R. Killip and B. Simon, Sum rules for Jacobi matrices and their applications to spectral theory, Annals of Math. (2) 158 (2003), 253321.
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 A. Laptev, S. Naboko, and O. Safronov, On new relations between spectral properties of Jacobi matrices and their coefficients, to appear.
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 S. Molchanov, M. Novitskii, and B. Vainberg, First KdV integrals and absolutely continuous spectrum for 1D Schrödinger operator, Comm. Math. Phys. 216 (2001), no. 1, 195213. MR 2001k:35259
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 O. Safronov, The spectral measure of a Jacobi matrix in terms of the Fourier transform of the perturbation, submitted.
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 G. Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, 72, Amer. Math. Soc., Providence, RI, 2000. MR 2001b:39019
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Additional Information
S. Kupin
Affiliation:
Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
Email:
kupin@math.brown.edu
DOI:
http://dx.doi.org/10.1090/S0002993903072447
PII:
S 00029939(03)072447
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
