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Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices

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Języki publikacji
EN
Abstrakty
EN
The research included in the paper concerns a class of symmetric block Jacobi matrices. The problem of the approximation of eigenvalues for a class of a self-adjoint unbounded operators is considered. We estimate the joint error of approximation for the eigenvalues, numbered from 1 to N, for a Jacobi matrix J by the eigenvalues of the finite submatrix J(n) of order pn x pn, where N = max{k ∈ N : k ≤ rpn} and r ∈ (0, 1) is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of J in the case p = 3.
Rocznik
Strony
311--330
Opis fizyczny
Bibliogr. 22 poz.
Twórcy
autor
  • AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Cracow, Poland, malejki@uci.agh.edu.pl
Bibliografia
  • [1] W. Arveson, C* -algebras and numerical linear algebra, J. Funct. Anal. 122 (1994) 2, 333–360.
  • [2] W. Arveson, Improper filtrations for C* -algebras: Spectra of unilateral tridiagonal operators, Acta Sci. Math. 57 (1993) 1–4, 11–24.
  • [3] A. Boutet de Monvel, S. Naboko, L. O. Silva, Eigenvalue asympthotics of a modified Jaynes-Cummings model with periodic modulations, C. R. Acad. Sci. Paris, Ser. I 338 (2004), 103–107.
  • [4] A. Boutet de Monvel, S. Naboko, L. Silva, The asympthotic behaviour of eigenvalues of modified Jaynes-Cummings model, Asymptot. Anal. 47 (2006) 3–4, 291–315.
  • [5] A. Boutet de Monvel, L. Zielinski, Eigenvalue asymptotics for Jaynes-Cummings type models without modulations, preprint 2008.
  • [6] P. Cojuhari, On the spectrun of a class of block Jacobi matrices, Operator Theory, Structured Matrices and Dilations, Theta (2007), 137–152.
  • [7] P. Cojuhari, J. Janas, Discreteness of the spectrum for some unbounded Jacobi matrices, Acta Sci. Math. (Szeged) 73 (2007), 649–667.
  • [8] H. Dette, B. Reuther, W.J. Studden, M. Zygmunt, Matrix measures and random walks with a block tridiagonal transition matrix, SIAM J. Matrix Anal. Appl. 29 (2006) 1, 117–142.
  • [9] J. Edward, Spectra of Jacobi matrices, differential equations on the circle, and the su(1, 1) Lie algebra, SIAM J. Math. Anal. 24 (1993) 3, 824–831.
  • [10] E.K. Ifantis, C.G. Kokologiannaki, E. Petropoulou, Limit points of eigenvalues of truncated unbounded tridiagonal operators, Cent. Eur. J. Math. 5 (2007) 2, 335–344.
  • [11] J. Janas, M. Malejki, Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices, J. Comput. Appl. Math. 200 (2007), 342–356.
  • [12] J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: Asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36 (2004) 2, 643–658.
  • [13] J. Janas, S. Naboko, Multithreshold spectral phase transitions for a class of Jacobi matrices, Operator Theory: Adv. Appl. 124 (2001), 267–285.
  • [14] H. Lutkepohl, Handbook of Matrices, John Wiley & Sons (1996).
  • [15] M. Malejki, Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l2 by the use of finite submatrices, Cent. Eur. J. Math. 8 (2010) 1, 114–128, DOI:10.2478/s11533-009-0064-x.
  • [16] M. Malejki, Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices, Opuscula Math. 27 (2007) 1, 37–49.
  • [17] M. Malejki, Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices, Linear Algebra and its Applications 431 (2009), 1952–1970.
  • [18] D. Masson, J. Repka, Spectral theory of Jacobi matrices in l2(Z) and the su(1, 1) Lie algebra, SIAM J. Math. Anal. 22 (1991), 1131–1146.
  • [19] Y. Saad, Numerical methods for large eigenvalue problems [in:] Algorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester, UK, 1992.
  • [20] G. Teschl, Jacobi operators and completely integrable nonlinear lattices, AMS Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2000.
  • [21] H. Volkmer, Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation, Constr. Approx. 20 (2004), 39–54.
  • [22] L. Zielinski, Eigenvalue asymptotics for a class of Jacobi matrices, Hot topics in operator theory, Theta Ser. Adv. Math., 9, Theta, Bucharest, 2008, 217–229.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHT-0003-0008
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