Spectral representation for singlet and triplet green's functions

Spectral representations of special Green functions are given explicitly. We consider the density correlation functionG ₂(x ₁ η ₁ x ₂ η ₂,x ₁ ⁺ η ₁ x ₂ ⁺ η′₂) and the functionG ₂(x ₁ η ₁ x ₁ ^? η₂,x ₂ η ₁ ^′ x ₂ ^? η ₂ ^′ Coupling the field operators Ψ^? (x, η), Ψ(x, η) to singlet and triplet operatorsA _{SMs TT3} (x), we obtain spectral representations for theseG-functions. The formulae derived may be of use when studying the system of equations for the Green functions, which describe many particle systems from a microscopic point of view.