Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem

Author:
Christian Remling

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2175-2182

MSC (2010):
Primary 42C05, 47B36, 81Q10

DOI:
https://doi.org/10.1090/S0002-9939-2010-10747-5

Published electronically:
November 30, 2010

MathSciNet review:
2775395

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If a Jacobi matrix is reflectionless on and has a single equal to , then is the free Jacobi matrix , . The paper discusses this result and its generalization to arbitrary sets and presents several applications, including the following: if a Jacobi matrix has some portion of its 's close to , then one assumption in the Denisov-Rakhmanov Theorem can be dropped.

**1.**Joseph A. Cima, Alec L. Matheson, and William T. Ross,*The Cauchy transform*, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. MR**2215991****2.**Walter Craig,*The trace formula for Schrödinger operators on the line*, Comm. Math. Phys.**126**(1989), no. 2, 379–407. MR**1027503****3.**D. Damanik, D. Hundertmark, R. Killip, and B. Simon,*Variational estimates for discrete Schrödinger operators with potentials of indefinite sign*, Comm. Math. Phys.**238**(2003), no. 3, 545–562. MR**1993385**, https://doi.org/10.1007/s00220-003-0868-7**4.**P. Deift and B. Simon,*Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension*, Comm. Math. Phys.**90**(1983), no. 3, 389–411. MR**719297****5.**Sergey A. Denisov,*On Rakhmanov’s theorem for Jacobi matrices*, Proc. Amer. Math. Soc.**132**(2004), no. 3, 847–852. MR**2019964**, https://doi.org/10.1090/S0002-9939-03-07157-0**6.**J. Dombrowski,*Quasitriangular matrices*, Proc. Amer. Math. Soc.**69**(1978), no. 1, 95–96. MR**0467373**, https://doi.org/10.1090/S0002-9939-1978-0467373-3**7.**S. Kotani,*Generalized Floquet theory for stationary Schrödinger operators in one dimension*, Chaos Solitons Fractals**8**(1997), no. 11, 1817–1854. MR**1477262**, https://doi.org/10.1016/S0960-0779(97)00042-8**8.**Alexei Poltoratski and Christian Remling,*Reflectionless Herglotz functions and Jacobi matrices*, Comm. Math. Phys.**288**(2009), no. 3, 1007–1021. MR**2504863**, https://doi.org/10.1007/s00220-008-0696-x**9.**A. Poltoratski and C. Remling, Approximation results for reflectionless Jacobi matrices, to appear in*Int. Math. Res. Notices*,`http://arxiv.org/abs/1005.2149`.**10.**E. A. Rakhmanov,*The asymptotic behavior of the ratio of orthogonal polynomials. II*, Mat. Sb. (N.S.)**118(160)**(1982), no. 1, 104–117, 143 (Russian). MR**654647****11.**C. Remling, The absolutely continuous spectrum of Jacobi matrices, to appear in*Annals of Math.***12.**Barry Simon,*Equilibrium measures and capacities in spectral theory*, Inverse Probl. Imaging**1**(2007), no. 4, 713–772. MR**2350223**, https://doi.org/10.3934/ipi.2007.1.713**13.**Barry Simon and Thomas Spencer,*Trace class perturbations and the absence of absolutely continuous spectra*, Comm. Math. Phys.**125**(1989), no. 1, 113–125. MR**1017742****14.**Gerald Teschl,*Jacobi operators and completely integrable nonlinear lattices*, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR**1711536**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
42C05,
47B36,
81Q10

Retrieve articles in all journals with MSC (2010): 42C05, 47B36, 81Q10

Additional Information

**Christian Remling**

Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019

Email:
cremling@math.ou.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10747-5

Keywords:
Reflectionless Jacobi matrix,
Denisov-Rakhmanov Theorem

Received by editor(s):
June 14, 2010

Published electronically:
November 30, 2010

Additional Notes:
The author’s work was supported by NSF grant DMS 0758594

Communicated by:
Walter Van Assche

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.