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

DOI:
https://doi.org/10.1090/S0002-9939-03-07244-7

Published electronically:
December 12, 2003

MathSciNet review:
2053342

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**1.**K. Case,*Orthogonal polynomials from the viewpoint of scattering theory*, J. Mathematical Phys.**15**(1974), 2166-2174. MR**50:6342****2.**K. Case,*Orthogonal polynomials, II*, J. Mathematical Phys.**16**(1975), 1435-1440. MR**56:2008****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**2001k:35259****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.**G. Teschl,*Jacobi operators and completely integrable nonlinear lattices*, Mathematical Surveys and Monographs, 72, Amer. Math. Soc., Providence, RI, 2000. MR**2001b:39019**

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

Email:
kupin@math.brown.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-07244-7

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