An Axiomatic Formulation of the Theory of Coinciding Simple Poles and Multiple Poles

Abstract

In the Bethe-Salpeter formalism, the scattering Green's function is known to have multiple poles synthesized out of coinciding simple poles. The present paper proposes an axiomatic approach to the problem of finding the residues of the multiple poles in terms of those of M coinciding simple poles. The latter residues are regarded as finite-dimensional, mutually orthogonal projection operators on a reflexive Banach space and its dual. Then various properties of the residues of the multiple poles are derived without recourse to the original Bethe-Salpeter equation, and especially it is shown mathematically that they can be decomposed into a direct sum of operators which commute with the Bethe-Salpeter operator. The residues of multiple poles are explicitly determined in two particular cases, M = N + 1 and M = 2, where N denotes the highest order of the singularities (in a parameter) of the residues of the coinciding simple poles.