  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 the system has the semirelativistic ``spinless-Salpeter''
Hamiltonian H = \sum_{i=1}^N \sqrt{m^2 + p_i^2} + \sum_{j>i=1}^N gamma |r_i -
r_j|^2, gamma > 0. We derive the following energy bounds: E(N) = min_{r>0} [N
(m^2 + 2 (N-1) P^2 / (N r^2))^1/2 + N (N-1) gamma r^2 / 2], N \ge 2, where
P=1.376 yields a lower bound and P=3/2 yields an upper bound for all N \ge 2. A
sharper lower bound is given by the function P = P(mu), where mu =
m(N/(gamma(N-1)^2))^(1/3), which makes the formula for E(2) exact: with this
choice of P, the bounds coincide for all N \ge 2 in the Schroedinger limit m
--> infinity.
