Asymptotic completeness for multi-particle schroedinger Hamiltonians with weak potentials

We show that the non-relativistic quantum mechanicaln-body HamiltoniansT(k)=T+kV andT, the free particle Hamiltonian, are unitarily equivalent in the center of mass system, i.e.,T(k)=W _± (k)TW _± (k) ^–1 fork sufficiently small and real. , a sum ofn(n–1)/2 real pair potentials,V _i, depending on the relative coordinatex _i R ³ of the pairi, whereV _i is required to behave like |x_i|^{– 2 –} as |x _i | and like |x_i|^{– 2 +} as |x _i |0.T(k) is the self-adjoint operator associated with the form sumT+kV. There are no smoothness requirements imposed on theV _i . Furthermore are the wave operators of time dependent scattering theory and are unitary. This result gives a quantitative form of the intuitive argument based on the Heisenberg uncertainty principle that a certain minimum potential well depth and range is needed before a bound state can be formed. This is the best possible long range behavior in the sense that ifkV _i C _i |x _i |^–b , 0<b2 for |x _i |>R _i (0<R _i <) and allC _i are negative thenT(k) has discrete eigenvalues andW _±(k) are not unitary.