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

Korotyaev, Evgeny; Kutsenko, Anton

doi:10.1090/S0002-9939-09-09827-X

Borg-type uniqueness theorems for periodic Jacobi operators with matrix-valued coefficients
HTML articles powered by AMS MathViewer

by Evgeny Korotyaev and Anton Kutsenko PDF

Proc. Amer. Math. Soc. 137 (2009), 1989-1996 Request permission

Abstract:

We give a simple proof of Borg-type uniqueness theorems for periodic Jacobi operators with matrix-valued coefficients.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
D. Bättig, B. Grébert, J.-C. Guillot, and T. Kappeler, Fibration of the phase space of the periodic Toda lattice, J. Math. Pures Appl. (9) 72 (1993), no. 6, 553–565. MR 1249409
Brüning, J.; Chelkak, D.; Korotyaev, E. Inverse spectral analysis for finite matrix-valued Jacobi operators, preprint, 2006.
Steve Clark and Fritz Gesztesy, On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems, J. Comput. Appl. Math. 171 (2004), no. 1-2, 151–184. MR 2077203, DOI 10.1016/j.cam.2004.01.011
Steve Clark, Fritz Gesztesy, and Walter Renger, 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, DOI 10.1016/j.jde.2005.04.013
H. Flaschka, Discrete and periodic illustrations of some aspects of the inverse method, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, pp. 441–466. MR 0455033
Fritz Gesztesy, Alexander Kiselev, and Konstantin A. Makarov, Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-type operators, Math. Nachr. 239/240 (2002), 103–145. MR 1905666, DOI 10.1002/1522-2616(200206)239:1<103::AID-MANA103>3.0.CO;2-F
E. Korotyaev, Gap-length mapping for periodic Jacobi matrices, Russ. J. Math. Phys. 13 (2006), no. 1, 64–69. MR 2262812, DOI 10.1134/S1061920806010067
Evgeni Korotyaev and Igor V. Krasovsky, Spectral estimates for periodic Jacobi matrices, Comm. Math. Phys. 234 (2003), no. 3, 517–532. MR 1964381, DOI 10.1007/s00220-002-0768-2
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.
Evgeny Korotyaev and Anton Kutsenko, Inverse problem for the discrete 1D Schrödinger operator with small periodic potentials, Comm. Math. Phys. 261 (2006), no. 3, 673–692. MR 2197543, DOI 10.1007/s00220-005-1429-z
E. Korotyaev and A. Kutsenko, Marchenko-Ostrovski mappings for periodic Jacobi matrices, Russ. J. Math. Phys. 14 (2007), no. 4, 448–452. MR 2366203, DOI 10.1134/S1061920807040115
Pierre van Moerbeke, The spectrum of Jacobi matrices, Invent. Math. 37 (1976), no. 1, 45–81. MR 650253, DOI 10.1007/BF01418827
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, vol. 72, American Mathematical Society, Providence, RI, 2000. MR 1711536, DOI 10.1090/surv/072