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

## TL;DR

This paper studies how to accurately approximate the spectra of singular discrete linear Hamiltonian systems with one singular endpoint using regular subspace extensions, providing convergence results and error estimates.

## Contribution

It establishes spectral approximation methods for singular Hamiltonian systems, proving spectral exactness in the limit circle case and spectral inclusion in other cases.

## Key findings

- Eigenvalues of regular approximations converge to those of the singular system.
- Spectral exactness holds in the limit circle case.
- Error estimates for eigenvalue approximations are provided.

## 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.