Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in \(l^{2}\) by the use of finite submatrices (Q2379255)

Let \(J\) be an infinite Jacobi matrix that defines a selfadjoint operator in \(l^{2}\) and \(J_{n}\) be the finite submatrices of order \(n\times n\). The author estimates the asymptotics (with \(n\to \infty\)) of the joint error of approximation for a part of eigenvalues of the operator \(J\) by the eigenvalues of \(J_{n}\). The method applied is based on \textit{H.\,Volkmer}'s results [Constructive Approximation 20, No.\,1, 39--54 (2004; Zbl 1063.33028)] and yields a description of the asymptotic behaviour of the point spectrum of \(J\). The approach to the problem is different from the methods applied by other authors.