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2014 | 12 | 3 | 445-463
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Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices

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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.
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Bibliografia
  • [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]
  • [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
  • [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] 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
  • [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
  • [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] 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
  • [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]
  • [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]
  • [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]
  • [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] 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]
  • [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]
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Bibliografia
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