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2010 | 8 | 1 | 114-128
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Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices

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