A lower bound for the spectrum of N-particle Hamiltonians

We consider the spectrum of an N-particle Hamiltonian H = Sigma (i) (1/2m(i)) Delta ((r) over right arrowi) + Sigma (i<j) V-ij(<(r)over right arrow> - (r) over right arrow (ji)) with translation-invariant pair interactions V-ij. H acts in L-2(R-3N). Letting sigma denote the spectrum we obtain the lower bound inf sigma (H) greater than or equal to Sigma (i<j) inf <sigma> (h(ij)') where h(ij)' = -(1/2 mu (ij)') Delta ((r) over right arrow) + V-ij((r) over right arrow), 1 less than or equal to i < j <less than or equal to> N is the single-pat-tide relative coordinate Hamiltonian with reduced mass mu (ij)' = (N - 1)mu (ij), mu (ij) = m(i)m(j) (m(i) + m(j))-1 acting in L-2(R-3). In particular, if sigma (h(ij)') subset of [0, infinity) (for example, weak pair interactions) for all i, j then H has no negative energy spectrum. For example, if each V-ij((r) over right arrow) is in L-3/2(R-3) and sufficiently small it is known that sigma (h(ij)') subset of [0, infinity).