A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems

TL;DR

This paper proves that for certain singular linear Hamiltonian systems, the analytic and geometric multiplicities of eigenvalues are equal, using a fundamental method applicable to both continuous and discrete cases with various endpoint conditions.

Contribution

It establishes the equality of eigenvalue multiplicities for a class of singular Hamiltonian systems, extending the understanding to both continuous and discrete scenarios with different endpoint behaviors.

Findings

Eigenvalue multiplicities are equal in singular Hamiltonian systems.

The proof applies to both continuous and discrete systems.

Method works for various endpoint regularity conditions.

Abstract

In this paper, we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear Hamiltonian systems are equal, where both endpoints are in the limit circle cases. The proof is fundamental and is given for both continuous and discrete Hamiltonian systems. The method used in this paper also works for both endpoints are regular, or one endpoint is regular and the other is in the limit circle case.