Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices
The present paper deals with some approaches to study the asymptotic behaviour of eigenvalues of a sufficiently large class of unbounded Jacobi matrices. In particular, the authors prove the following result (Theorem 2.2). Let \(J\) be an Jacobi operator defined on its maximal domain in \(l_{2} (\mathbb{N})\) by \(J = S W + W S^{*} + Q,\) where \(Q = \text{diag} (q_n)\), \(W = \text{diag} (\lambda_{n})\) (\((q_n)\) and \((\lambda_n)\) are given sequences of positive numbers), \(S\) is the canonical shift operator and assume that (i) \(q_n = \rho n^{\alpha} (1 + \Delta_{n})\), \(\Delta_{n} \longrightarrow 0\), \(\rho \in \mathbb{R}\;\backslash \{0\}\); (ii) \(\beta \geq 0\), \(\alpha > 2 \beta + 1\); (iii) \(\Delta_{n + 1} - \Delta_{n} = O(1/n)\). Then the eigenvalues \(\lambda_{n} (J)\) of \(J\) satisfy \(\lambda_{n} (J) = q_{n} + O (1/n^{\alpha-2\beta-1})\) for large \(n\).
scientific article; zbMATH DE number 5117689
scientific article