Opuscula Math. 30, no. 3 (2010), 311-330

Opuscula Mathematica

Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices

Maria Malejki

Abstract. 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 \(pn \times pn\), where \(N = \max \{k \in \mathbb{N}: k \leq rpn\}\) and \(r \in (0,1)\) is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of \(J\) in the case \(p=3\).

Keywords: symmetric unbounded Jacobi matrix, block Jacobi matrix, tridiagonal matrix, point spectrum, eigenvalue, asymptotics.

Mathematics Subject Classification: 47A75, 47B25, 47B36, 15A18.

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  • Maria Malejki
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland
  • Received: 2010-02-19.
  • Revised: 2010-04-26.
  • Accepted: 2010-04-30.
Opuscula Mathematica - cover

Cite this article as:
Maria Malejki, Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices, Opuscula Math. 30, no. 3 (2010), 311-330, http://dx.doi.org/10.7494/OpMath.2010.30.3.311

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