Scattering matrices and generalized Fourier transforms in long-range $N$-body problems

We give a definition of scattering matrices based on the asymptotic behaviors of generalized eigenfunctions and show that these scattering matrices are equivalent to the ones defined by wave-operator approach in long-range $N$-body problems. We also define generalized Fourier transforms by the asymptotic behaviors of outgoing solutions to nonhomogeneous equations and show that they are equivalent to the definition using wave operators. We also prove that the adjoint operators of the generalized Fourier transforms are given by Poisson operators.