On the infrared singularity of the resolvent of some Yang-Mills-type operators

For operators of Yang-Mills type H = Σ_{j = 1}⁴ [ -i(∂/∂cursive Greek chi_j) ⊗ 1 + A _j(cursive Greek chi)]² + W (cursive Greek chi) in L ²(R⁴) ⊗ Mathematical script capital V the infrared singularity of the resolvent (H - ζ)^-1 is described completely in terms of asymptotic expansions as ξ→0.A_j(cursive Greek chi) and W(cursive Greek chi) are required to satisfy A_j(cursive Greek chi) = O (\cursive Greek chi\^-2-δ), W(cursive Greek chi) = O(\cursive Greek chi\^-2-δ), as\cursive Greek chi\→∞, δ > 0.