Modified Lippmann-Schwinger equations for two-body scattering theory with long-range interactions

Abstract

Two kinds of modified Lippmann‐Schwinger equations are derived for the case of long‐range potentials. The equations of the first kind are homogeneous and are a direct result of the fact that the standard Lippmann‐Schwinger equations do not hold when long‐range forces are present. The equations of the second kind depend on the existence of an operator Z such that W_± = s‐lim exp (iHt)Z exp(−iH₀t). A general recipe for constructing Z is given and its computation is carried through for the case of asymptotically Coulombic potentials. The resulting equations are used to compare the long‐range theory with the theory with a space cutoff (i.e., screened potential) in the limit in which that cutoff is being removed.