Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices

Maria Malejki

Open Mathematics (2010)

  • Volume: 8, Issue: 1, page 114-128
  • ISSN: 2391-5455

Abstract

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We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].

How to cite

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Maria Malejki. "Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices." Open Mathematics 8.1 (2010): 114-128. <http://eudml.org/doc/269506>.

@article{MariaMalejki2010,
abstract = {We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max\{k ∈ ℕ: k ≤ rn\} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].},
author = {Maria Malejki},
journal = {Open Mathematics},
keywords = {Self-adjoint unbounded Jacobi matrix; Asymptotics; Point spectrum; Tridiagonal matrix; Eigenvalue; selfadjoint unbounded Jacobi matrix; asymptotics; point spectrum; tridiagonal matrix; eigenvalues},
language = {eng},
number = {1},
pages = {114-128},
title = {Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices},
url = {http://eudml.org/doc/269506},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Maria Malejki
TI - Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 114
EP - 128
AB - We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].
LA - eng
KW - Self-adjoint unbounded Jacobi matrix; Asymptotics; Point spectrum; Tridiagonal matrix; Eigenvalue; selfadjoint unbounded Jacobi matrix; asymptotics; point spectrum; tridiagonal matrix; eigenvalues
UR - http://eudml.org/doc/269506
ER -

References

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  1. [1] Arveson W., C *-algebras and numerical linear algebra, J. Funct. Anal., 1994, 122(2), 333–360 http://dx.doi.org/10.1006/jfan.1994.1072 
  2. [2] Arveson W., Improper filtrations for C *-algebras: Spectra of unilateral tridiagonal operators, Acta Sci. Math. (Szeged), 1993, 57(1–4), 11–24 Zbl0819.46044
  3. [3] Bonini M., Cicuta G.M., Onofri E., Fock space methods and large N, J. Phys. A, 2007, 40(10), F229–F234 http://dx.doi.org/10.1088/1751-8113/40/10/F01 Zbl1109.81085
  4. [4] Boutet de Monvel A., Naboko S., Silva L., The asympthotic behaviour of eigenvalues of modified Jaynes-Cummings model, Asymptot. Anal., 2006, 47(3–4), 291–315 Zbl1139.47024
  5. [5] Boutet de Monvel A., Naboko S., Silva L.O., Eigenvalue asympthotics of a modified Jaynes-Cummings model with periodic modulations, C. R. Math. Acad. Sci. Paris, Ser. I, 2004, 338, 103–107 Zbl1037.47019
  6. [6] Boutet de Monvel A., Zielinski L., Eigenvalue asymptotics for Jaynes-Cummings type models without modulations, preprint 
  7. [7] Cojuhari P., Janas J., Discreteness of the spectrum for some unbounded Jacobi matrices, Acta Sci. Math. (Szeged), 2007, 73, 649–667 Zbl1211.47059
  8. [8] Djakov P., Mityagin B., Simple and double eigenvalues of the Hill operator with a two term potential, J. Approx. Theory, 2005, 135(1), 70–104 http://dx.doi.org/10.1016/j.jat.2005.03.004 Zbl1080.34066
  9. [9] Edward J., Spectra of Jacobi matrices, differential equations on the circle, and the su(1; 1) Lie algebra, SIAM J. Math. Anal., 1993, 24(3), 824–831 http://dx.doi.org/10.1137/0524051 Zbl0777.39003
  10. [10] Kozhan R.V., Asymptotics of the eigenvalues of two-diagonal Jacobi matrices, Mathematical Notes, 2005, 77(2), 283–287 http://dx.doi.org/10.1007/s11006-005-0028-9 Zbl1082.15021
  11. [11] Ifantis E.K., Kokologiannaki C.G., Petropoulou E., Limit points of eigenvalues of truncated unbounded tridiagonal operators, Cent. Eur. J. Math., 2007, 5(2), 335–344 http://dx.doi.org/10.2478/s11533-007-0009-1 Zbl1130.47002
  12. [12] Janas J., Malejki M., Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices, J. Comput. Appl. Math., 2007, 200, 342–356 http://dx.doi.org/10.1016/j.cam.2005.09.033 Zbl1110.47020
  13. [13] Janas J., Naboko S., Multithreshold Spectral Phase Transitions for a Class of Jacobi Matrices, Oper. Theory Adv. Appl., 2001, 124, 267–285 Zbl1017.47025
  14. [14] 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 Zbl1091.47026
  15. [15] Malejki M., Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices, Opuscula Math., 2007, 27/1, 37–49 Zbl1155.47034
  16. [16] Malejki M., Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices, Linear Algebra Appl., 2009, 431, 1952–1970 http://dx.doi.org/10.1016/j.laa.2009.06.035 
  17. [17] Masson D., Repka J., Spectral theory of Jacobi matrices in l 2(ℤ) and the su(1; 1) Lie algebra, SIAM J. Math. Anal., 1991, 22, 1131–1146 http://dx.doi.org/10.1137/0522073 Zbl0729.33011
  18. [18] Rozenbljum G.V., Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, Tr. Mosk. Mat. Obs., 1978, 36, 59–84 (in Russian) 
  19. [19] Saad Y., Numerical methods for large eigenvalue problems, In: Algorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester, UK, 1992 
  20. [20] Shivakumar P.N., Williams J.J., Rudraiah N., Eigenvalues for infinite matrices, Linear Algebra Appl., 1987, 96, 35–63 http://dx.doi.org/10.1016/0024-3795(87)90335-1 
  21. [21] Tur E.A., Jaynes-Cummings model: Solution without rotating wave approximation, Optics and Spectroscopy, 2000, 89(4), 574–588 http://dx.doi.org/10.1134/1.1319919 
  22. [22] Tur E.A., Asymptotics of eigenvalues for a class of Jacobi matrices with limiting point spectrum, Mathematical Notes, 2003, 74(3), 425–437 http://dx.doi.org/10.1023/A:1026171122104 Zbl1081.47037
  23. [23] Volkmer H., Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation, Constr. Approx., 2004, 20, 39–54 http://dx.doi.org/10.1007/s00365-002-0527-9 Zbl1063.33028
  24. [24] Wilkinson J.H., Rigorous error bounds for coputed eigensystems, Comput. J., 1961, 4, 230–241 http://dx.doi.org/10.1093/comjnl/4.3.230 Zbl0109.34504
  25. [25] Zielinski L., Eigenvalue asymptotics for a class of Jacobi matrices, Hot topics in operator theory, Theta Ser. Adv. Math., 9, Theta, Bucharest, 2008, 217–229 Zbl1199.47142

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