Browsing by Subject "block Jacobi matrix"

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Access status: Open Access ,
Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices
(2010) Malejki, Maria
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 $p = 3$.
Access status: Open Access ,
Block Jacobi matrices and Titchmarsh-Weyl function
(Wydawnictwa AGH, 2025) Moszyński, Marcin; Świderski, Grzegorz
We collect some results and notions concerning generalizations for block Jacobi matrices of several concepts, which have been important for spectral studies of the simpler and better known scalar Jacobi case. We focus here on some issues related to the matrix Titchmarsh-Weyl function, but we also consider generalizations of some other tools used by subordinacy theory, including the matrix orthogonal polynomials, the notion of finite cyclicity, a variant of a notion of nonsubordinacy, as well as Jitomirskaya-Last type semi-norms. The article brings together some issues already known, our new concepts, and also improvements and strengthening of some results already existing. We give simpler proofs of some known facts, or we add details usually omitted in the existing literature. The introduction contains a separate part devoted to a brief review of the main spectral analysis methods used so far for block Jacobi operators.