Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices

Abstract

We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space \(l^2(\mathbb{N})\) by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with \(n\to \infty\)) of the joint error of approximation for the first \(n\) eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order \(n \times n\). The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [H. Volkmer, Error Estimates for Rayleigh-Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation, Constr. Approx. 20 (2004), 39-54]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.

Keywords

• self-adjoint unbounded Jacobi matrix
• asymptotics
• point spectrum
• tridiagonal matrix
• eigenvalue