Regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint?

Abstract

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of induced regular SSEs, generated by the same difference expression on smaller finite intervals. It is shown that every SSE of the minimal subspace has a pure discrete spectrum, and the k-th eigenvalue of any given SSE is exactly the limit of the k-th eigenvalues of the induced regular SSEs; that is, spectral exactness holds, in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, in the limit point and intermediate cases, spectral inclusive holds.