  We develop error estimates for the semi-discrete conservative spectral method
for the approximation of the elastic and inelastic space homogeneous Boltzmann
equation introduced by the authors in \cite{GT09}. In addition we study the
long time convergence of such semi-discrete solution to equilibrium Maxwellian
distribution that conserves the mass, momentum and energy associated to the
initial data. The numerical method is based on the Fourier transform of the
collisional operator and a Lagrangian optimization correction that enforces the
collision invariants, namely conservation of mass, momentum and energy in the
elastic case, and just mass and momentum in the inelastic one. We present a
detailed semi-discrete analysis on convergence of the proposed numerical method
which includes the $L^{1}-L^{2}$ theory for the scheme. This analysis allows us
to present, additionally, convergence in Sobolev spaces and convergence to
equilibrium for the numerical approximation. The results of this work answer a
long standing open problem posed by Cercignani et al. in \cite[Chapter 12]{CIP}
about finding error estimates for a numerical scheme associated to the
Boltzmann equation, as well as showing the semi-discrete numerical solution
converges to the equilibrium Maxwellian distribution associated to the initial
value problem.
