  We present a semiclassical study of the spectrum of a few-body system
consisting of two short-range interacting bosonic particles in one dimension, a
particular case of a general class of integrable many-body systems where the
energy spectrum is given by the solution of algebraic transcendental equations.
By an exact mapping between $\delta$-potentials and boundary conditions on the
few-body wave functions, we are able to extend previous semiclassical results
for single-particle systems with mixed boundary conditions to the two-body
problem. The semiclassical approach allows us to derive explicit analytical
results for the smooth part of the two-body density of states that are in
excellent agreement with numerical calculations. It further enables us to
include the effect of bound states in the attractive case. Remarkably, for the
particular case of two particles in one dimension, the discrete energy levels
obtained through a requantization condition of the smooth density of states are
essentially in perfect agreement with the exact ones.
