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Borg-type uniqueness theorems for periodic Jacobi operators with matrix-valued coefficients


Authors: Evgeny Korotyaev and Anton Kutsenko
Journal: Proc. Amer. Math. Soc. 137 (2009), 1989-1996
MSC (2000): Primary 47B39, 34A55, 47B36
DOI: https://doi.org/10.1090/S0002-9939-09-09827-X
Published electronically: January 29, 2009
MathSciNet review: 2480280
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Abstract: We give a simple proof of Borg-type uniqueness theorems for periodic Jacobi operators with matrix-valued coefficients.


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  • [AS] Abramowitz, M.; Stegun, A., eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications Inc., New York, 1992. MR 1225604 (94b:00012)
  • [BGGK] Bättig, D.; Grébert, B.; Guillot, J.-C.; Kappeler, T. Fibration of the phase space of the periodic Toda lattice. J. Math. Pures Appl. (9) 72 (1993), no. 6, 553-565. MR 1249409 (95b:58073)
  • [BCK] Brüning, J.; Chelkak, D.; Korotyaev, E. Inverse spectral analysis for finite matrix-valued Jacobi operators, preprint, 2006.
  • [CG] Clark, S.; Gesztesy, F. On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems. J. Comput. Appl. Math. 171 (2004), 151-184. MR 2077203 (2006i:39033)
  • [CGR] Clark, S.; Gesztesy, F.; Renger, W. Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators. J. Differential Equations 219 (2005), no. 1, 144-182. MR 2181033 (2006e:47068)
  • [F] Flaschka, H. Discrete and periodic illustrations of some aspects of the inverse method, in Dynamical Systems, Theory and Applications, J. Moser (ed.), Lecture Notes In Physics, vol. 38, Springer, Berlin, 1975, pp. 441-466. MR 0455033 (56:13274)
  • [GKM] Gesztesy, F.; Kiselev, A.; Makarov, K. A. Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-type operators. Math. Nachr. 239-240 (2002), 103-145. MR 1905666 (2003i:47047)
  • [K] Korotyaev, E. Gap-length mapping for periodic Jacobi matrices. Russ. J. Math. Phys. 13 (2006), no. 1, 64-69. MR 2262812 (2007g:39028)
  • [KKr] Korotyaev, E.; Krasovsky, I. Spectral estimates for periodic Jacobi matrices. Commun. Math. Phys. 234 (2003), 517-532. MR 1964381 (2003m:47056)
  • [KKu] Korotyaev, E.; Kutsenko, A. Lyapunov functions for periodic matrix-valued Jacobi operators, to appear in ``Spectral Theory of Differential Operators: M. Sh. Birman 80th Anniversary Collection'', American Mathematical Society Translations - Series 2, vol. 225, Amer. Math. Soc., Providence, RI, 2009.
  • [KKu1] Korotyaev, E.; Kutsenko, A. Inverse problem for the discrete 1D Schrödinger operator with small periodic potentials. Commun. Math. Phys. 261 (2006), 673-692. MR 2197543 (2006j:81061)
  • [KKu2] Korotyaev, E.; Kutsenko, A. Marchenko-Ostrovski mappings for periodic Jacobi matrices. Russ. J. Math. Phys. 14 (2007), no. 4, 448-452. MR 2366203 (2008m:47043)
  • [vM] van Moerbeke, P. The spectrum of Jacobi matrices. Invent. Math. 37 (1976), no. 1, 45-81. MR 0650253 (58:31226)
  • [RS] Reed, M.; Simon, B. Methods of Modern Mathematical Physics, Vol. IV, Analysis of Operators, Academic Press, New York-London, 1978. MR 0493421 (58:12429c)
  • [T] Teschl, G. Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, vol. 72, Amer. Math. Soc., Providence, RI, 2000. MR 1711536 (2001b:39019)

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Additional Information

Evgeny Korotyaev
Affiliation: School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, United Kingdom
Email: KorotyaevE@cf.ac.uk

Anton Kutsenko
Affiliation: Department of Mathematics, Saint Petersburg State University, Saint Petersburg, 199034, Russia
Email: kucenkoa@rambler.ru

DOI: https://doi.org/10.1090/S0002-9939-09-09827-X
Received by editor(s): January 23, 2008
Published electronically: January 29, 2009
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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