FREE CHANNEL FOURIER TRANSFORM IN THE LONG-RANGE N-BODY PROBLEM By IRA HERBST* AND ERIK SKIBSTED 1. Introduction With the recent progress on the problem of asymptotic completeness [DI,SS 1 ] (see also [G] and [SS2]), it is natural to turn to another important problem in the study of N-particle scattering: the explicit construction of continuum eigen- functions and the distorted Fourier transform which diagonalizes the Hamiltonian. Asymptotic completeness is now known [D 1] for a nice class of long-range N-body Hamiltonians (with two-body potentials which decay faster than r -u, # = v~- 1). In this work we consider a class of long-range N-body operators with two-body potentials decaying faster than r -~ for some E > 0. We construct the free channel generalized eigenfunctions directly from the Green's function as was done by Agmon [A] in the two-body problem. This is accomplished using strong propagation estimates [Sk] and methods similar to those developed in [HS] for the two-body problem. Our methods also allow us to prove smoothness in the momenta and growth properties in configuration space of the generalized eigenfunctions. After the construction of the eigenfunctions, the next step is to show that they actually span the free channel (i.e., the range of a certain wave operator) and to identify the distorted Fourier transform. We state our results in Section 2. The remainder of the paper is spent proving them. It is interesting that our work is in some sense independent of the problem of asymptotic completeness. In particular, even if no k-body subsystem (2 < k < N - 1) has a bound state, our results do not yield asymptotic completeness which in this case is the statement that the range of the free channel wave operators is the continuous spectral subspace. Rather our results give a diagonalization of H when restricted to the range of the free channel wave operators. 2. Results We first state our assumptions and establish notation. In addition, we define and discuss the auxiliary quantities which we will need to frame our results. We ~ Researchsupportedby NSF grant DMS-8807816. 297 JOURNAL D'ANALYSE MATH15.MATIQU E, Vol. 65 ( 1995 )