Completeness in the sense of Cauchy principal value of Eigensystems for infinite dimensional Hamiltonian operators

Abstract

This paper studies the completeness properties of eigensystems associated with a class of infinite dimensional Hamiltonian operators (IDHOs). We establish necessary and sufficient conditions for the completeness in the sense of the Cauchy Principal Value (CCPV) for the eigensystems of these operators. Some examples are provided to illustrate the validity of the criteria. Additionally, we provide sufficient conditions for the CCPV of the eigensystems for specific classes of $4\times 4$ IDHOs.