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We solve the inverse scattering problem for the nonlinear Schrödinger equation on R n , n ≥ 3: We prove that the small-amplitude limit of the scattering operator uniquely determines V j , j = 0, 1, · · · . Our proof gives a method for the reconstruction of the potentials V j , j = 0, 1, · · · . The results of this paper extend our previous results for the problem on the line.doi:10.1090/s0002-9939-01-06016-6 fatcat:jin4c55qyrdvtiawklsvchkhw4