  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.
