Asymptotic completeness for multi-particle schroedinger Hamiltonians with weak potentials

Abstract

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.$V = \sum\limits_i {V_i }$, 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 −ε} as |x _i |→∞ and like |x_i|^{− 2 +ε} 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$W_ \pm (k) = \mathop {s - \lim }\limits_{t \to \pm \infty } e^{iT(k)t} e^{ - iTt}$ 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<b≦2 for |x _i |>R _i (0<R _i <∞) and allC _i are negative thenT(k) has discrete eigenvalues andW _±(k) are not unitary.