  We present a general method for deriving the energy shift of an interacting
system of $N$ spinless particles in a finite volume. To this end, we use the
nonrelativistic effective field theory (NREFT), and match the pertinent
low-energy constants to the scattering amplitudes. Relativistic corrections are
explicitly included up to a given order in the $1/L$ expansion. We apply this
method to obtain the ground state of $N$ particles, and the first excited state
of two and three particles to order $L^{-6}$ in terms of the threshold
parameters of the two- and three-particle relativistic scattering amplitudes.
We use these expressions to analyze the $N$-particle ground state energy shift
in the complex $\varphi^4$ theory.
