Relativistic N-boson systems bound by oscillator pair potentials

We study the lowest energy E of a relativistic system of N identical bosons bound by harmonic-oscillator pair potentials in three spatial dimensions. In natural units (ℎ/2π)=c=1 the system has the semirelativistic (or 'spinless-Salpeter') Hamiltonian, H=Σi=1N√(m2+pi2)+Σj>i=1Nγ vertical bar ri-rj vertical bar2, γ>0. We derive the following energy bounds: E(N)=minr>0[N(m2+2(N-1)P2/(Nr2))1/2+(N/2)(N-1)γr2], N≥2, where P=1.376 yields a lower bound and P=(3/2) yields an upper bound for all N≥2. A sharper lower bound is given by the function P=P(m) which makes the formula for E(2) exact: with this choice of P, the bounds coincide for all N≥2 in the Schroedinger limit defined by m→∞