Regular approximation of linear Hamiltonian operators with two singular endpoints

Abstract

In this paper we investigate regular approximation of linear Hamiltonian operators with two singular endpoints. For any given singular self-adjoint Hamiltonian operator, its spectrum can be approximated with the eigenvalues of a sequence of regular operators. The inherited restriction operators are constructed by the limit-circle solutions and the limit-point solutions, and proved to be self-adjoint. Via the abstract operator theory, we obtain the strongly resolvent convergence of the inherited regular operators and spectral inclusion of spectrum. Moreover for the case with both endpoints being limit circle, spectral exactness of spectrum is obtained by Hilbert-Schmidt norm convergence. In the case of Friedrichs extension of singular Hamiltonian systems, spectral exactness also holds due to variational principle of eigenvalues.