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In the hyperradial adiabatic (HA) treatment of the three-body  problem the total wave function is i expanded as ΨHA(R, ξ, η) = R−5/2 ∑iχi(R)φi(R|ξ, η),where R denotes the hyperradius  and (ξ , η) are internal hyperangles. Integration over  ξ and η converts the Schrödinger equation  into a system of coupled hyperradial equations. It is a well-known fact that,  within the HA approach, the non-adiabatic corrections that  couple channels converging to the same asymptotic conguration can show an  unphysical long-range behavior ∼ 1/R. Though the latter is of  purely kinematic origin and arises from the use of the hyperradius instead of  the pertinent Jacobi variables, it is nevertheless the source of the  considerable diculties inherent in this approach. Here we propose, following  the analysis of [1, 2], to dene appropriate hyperradial-distorted free incoming  and outgoing waves (H DF W ) that incorporate these unphysical long-range  eects. Using them the physical S−matrix can be found in a  straightforward manner

Deﬁnition of an appropriate free dynamics and the physicalS-matrix in multichannel hyperradial adiabatic scattering