  The spectrum of the Lipkin-Meshkov-Glick model is exactly derived in the
thermodynamic limit by means of a spin coherent states formalism. In the first
step, a classical analysis allows one to distinguish between four distinct
regions in the parameter space according to the nature of the singularities
arising in the classical energy surface; these correspond to spectral critical
points. The eigenfunctions are then analyzed more precisely in terms of the
associated roots of the Majorana polynomial, leading to exact expressions for
the density of states in the thermodynamic limit. Finite-size effects are also
analyzed, leading in particular to logarithmic corrections near the
singularities occuring in the spectrum. Finally, we also compute expectation
values of the spin operators in a semi-classical analysis in order to
illustrate some subtle effects occuring in one region of the parameter space.
