For most quantum mechanical systems of physical interest, central properties like the energy spectrum are not exactly accessable by analytical methods. In single-particle systems, however, many of the quantum features can be analytically adressed in semiclassical approximations that have been developed over the last decades and are now well established and sophisticated.

This work presents the transfer of semiclassical methods to systems of many identical and indistinguishable particles, showing the conceptual differences to single-particle systems. The concepts of the counterpart of periodic orbit theory are adressed and a classical sum rule is developed as a many-body-extension of a similar sum rule for single-particle systems, which is an useful tool for estimating statistical properties of many-body energy spectra.

The central result is a novel analytical approach to the calculation of the mean density of states in many-body billiard systems. The presented method makes explicit the intrinsic geometry inherent in the symmetrization postulate and, in the spirit of the usual Weyl expansion for the smooth part of the density of states in single-particle confined systems, the result takes the form of a sum over clusters of particles moving freely around manifolds in configuration space invariant under elements of the group of permutations.
Being asymptotic, the approximation gives increasingly better results for large excitation energies and comparison shows that it coincides with the celebrated Bethe estimate in the appropriate region. Moreover, the construction gives the correct high energy asymptotics expected from general considerations, and shows that the emergence of the fermionic ground state is actually a consequence of an extremely delicate large cancellation effect.
Remarkably, the expansion in cluster zones is naturally incorporated for systems of interacting particles, opening the road to address the fundamental problem about the interplay between exchange symmetry and interactions in many-body systems of identical particles.