Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices

Anne Monvel; Lech Zielinski

Open Mathematics (2014)

  • Volume: 12, Issue: 3, page 445-463
  • ISSN: 2391-5455

Abstract

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We consider an infinite Jacobi matrix with off-diagonal entries dominated by the diagonal entries going to infinity. The corresponding self-adjoint operator J has discrete spectrum and our purpose is to present results on the approximation of eigenvalues of J by eigenvalues of its finite submatrices.

How to cite

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Anne Monvel, and Lech Zielinski. "Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices." Open Mathematics 12.3 (2014): 445-463. <http://eudml.org/doc/269580>.

@article{AnneMonvel2014,
abstract = {We consider an infinite Jacobi matrix with off-diagonal entries dominated by the diagonal entries going to infinity. The corresponding self-adjoint operator J has discrete spectrum and our purpose is to present results on the approximation of eigenvalues of J by eigenvalues of its finite submatrices.},
author = {Anne Monvel, Lech Zielinski},
journal = {Open Mathematics},
keywords = {Jacobi matrices; Eigenvalue estimates; Error estimates; Helffer-Sjöstrand formula; eigenvalue estimates; error estimates},
language = {eng},
number = {3},
pages = {445-463},
title = {Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices},
url = {http://eudml.org/doc/269580},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Anne Monvel
AU - Lech Zielinski
TI - Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices
JO - Open Mathematics
PY - 2014
VL - 12
IS - 3
SP - 445
EP - 463
AB - We consider an infinite Jacobi matrix with off-diagonal entries dominated by the diagonal entries going to infinity. The corresponding self-adjoint operator J has discrete spectrum and our purpose is to present results on the approximation of eigenvalues of J by eigenvalues of its finite submatrices.
LA - eng
KW - Jacobi matrices; Eigenvalue estimates; Error estimates; Helffer-Sjöstrand formula; eigenvalue estimates; error estimates
UR - http://eudml.org/doc/269580
ER -

References

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  1. [1] Boutet de Monvel A., Naboko S., Silva L.O., Eigenvalue asymptotics of a modified Jaynes-Cummings model with periodic modulations, C. R. Math. Acad. Sci. Paris, 2004, 338(1), 103–107 http://dx.doi.org/10.1016/j.crma.2003.12.001[Crossref] Zbl1037.47019
  2. [2] Boutet de Monvel A., Naboko S., Silva L.O., The asymptotic behavior of eigenvalues of a modified Jaynes-Cummings model, Asymptot. Anal., 2006, 47(3–4), 291–315 Zbl1139.47024
  3. [3] Boutet de Monvel A., Zielinski L., Explicit error estimates for eigenvalues of some unbounded Jacobi matrices, In: Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, Berlin, July 12–16, 2010, Oper. Theory Adv. Appl., 221, Birkhäuser/Springer, Basel, 2012, 187–217 
  4. [4] Boutet de Monvel A., Zielinski L., Asymptotic behavior of large eigenvalues of a modified Jaynes-Cummings model, preprint available at http://www.math.uni-bielefeld.de/~bibos/preprints/12-07-409.pdf Zbl1320.47032
  5. [5] Cojuhari P.A., Janas J., Discreteness of the spectrum for some unbounded Jacobi matrices, Acta Sci. Math. (Szeged), 2007, 73(3–4), 649–667 Zbl1211.47059
  6. [6] Davies E.B., Spectral Theory and Differential Operators, Cambridge Stud. Adv. Math., 42, Cambridge University Press, Cambridge, 1995 http://dx.doi.org/10.1017/CBO9780511623721[Crossref] 
  7. [7] Helffer B., Sjöstrand J., Équation de Schrödinger avec champ magnétique et équation de Harper, In: Schrödinger Operators, Sønderborg, August 1–12, 1988, Lecture Notes in Phys., 345, Springer, Berlin, 1989, 118–197 Zbl0714.34130
  8. [8] Janas J., Malejki M., Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices, J. Comput. Appl. Math., 2007, 200(1), 342–356 http://dx.doi.org/10.1016/j.cam.2005.09.033[WoS][Crossref] Zbl1110.47020
  9. [9] Janas J., Naboko S., Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal., 2004, 36(2), 643–658 http://dx.doi.org/10.1137/S0036141002406072[Crossref] Zbl1091.47026
  10. [10] Janas J., Naboko S., Stolz G., Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices, Int. Math. Res. Not. IMRN, 2009, 4, 736–764 [WoS] Zbl1175.47029
  11. [11] Malejki M., Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices, Linear Algebra Appl., 2009, 431(10), 1952–1970 http://dx.doi.org/10.1016/j.laa.2009.06.035[Crossref][WoS] 
  12. [12] Malejki M., Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices, Cent. Eur. J. Math., 2010, 8(1), 114–128 http://dx.doi.org/10.2478/s11533-009-0064-x[WoS] Zbl1197.47046
  13. [13] Volkmer H., Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation, Constr. Approx., 2004, 20(1), 39–54 http://dx.doi.org/10.1007/s00365-002-0527-9[Crossref] Zbl1063.33028

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