The ground state energy of a classical gas

In this paper we study the ground state energy of a classical gas. Our interest centers mainly on Coulomb systems. We obtain some new lower bounds for the energy of a Coulomb gas. As a corollary of our results we can show that a fermionic system with relativistic kinetic energy and Coulomb interaction is stable. More precisely, letH _N (α) be theN particle Hamiltonian $$H_N (alpha ) = alpha sumlimits_{i = 1}^N {( - Delta _i )^{1/2} + } sumlimits_{i< j} {left| {x_i - x_j } right|^{ - 1} } - sumlimits_{i,j} {left| {x_i - R_j } right|^{ - 1} } + sumlimits_{i< j} {left| {R_i - R_j } right|^{ - 1} } $$ where Δ _i is the Laplacian in the variablex _i ∈?³ andR ₁, ...,R _N are fixed points in ?³. We show that for sufficiently large α, independent ofN, the HamiltonianH _N (α) is nonnegative on the space of square integrable functions ψ(x ₁, ...,x _N ), antisymmetric in the variablesx _i , 1≦i≦N.