Opuscula Math. 30, no. 3 (2010), 311-330
http://dx.doi.org/10.7494/OpMath.2010.30.3.311

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.

Full text (pdf)