Absence of Negative Energy Spectrum for N-Particle Hamiltonians

We consider the spectrum of the quantum Hamiltonian H for a system of N one-dimensional particles. H is given by $H = sumnolimits_{i = 1}^n { - frac{1}{{2m_i }}frac{{partial ^2 }}{{partial x_i^2 }}} + sum {_{1 leqslant i < j leqslant N} } V_{ij} left( {x_i - x_j } right)$ acting in L ²(R ^N ). We assume that each pair potential is a sum of a hard core for |x|≤a, a>0, and a function V _ij (x), |x|>a, with $smallint _a^infty left| {x - a} right|left| {V_{ij} left( x right)} right|dx < infty $ . We give conditions on V ^? _ij (x), the negative part of V _ij (x), which imply that H has no negative energy spectrum for all N. For example, this is the case if V ^? _ij (x) has finite range 2a and $$2m_i smallint _a^{2a} left| {x - a} right|left| {V_{ij}^ - left( x right)} right|dx < 1.$$ If V ^? _ij is not necessarily small we also obtain a thermodynamic stability bound inf?σ(H)≥?cN, where 0<c<∞, is an N-independent constant.