  We present a spectral Petrov-Galerkin method for the Boltzmann collision
operator. We expand the density distribution $f$ to high order orthogonal
polynomials multiplied by a Maxwellian. By that choice, we can approximate on
the whole momentum domain $\mathbb{R}^3$ resulting in high accuracy at the
evaluation of the collision operator. Additionally, the special choice of the
test space naturally ensures conservation of mass, momentum and energy. By
numerical examples we demonstrate the convergence (w.r.t. time) to the exact
stationary solution. For efficiency we transfer between nodal and Maxwellian
weighted Spherical Harmonics which are orthogonal w.r.t. the innermost
integrals of the collision operator. Combined with efficient transformations
between the bases and the calculation of the outer integrals this gives an
algorithm of complexity $\mathcal{O}(N^7)$ and a storage requirement
$\mathcal{O}(N^4)$ for the evaluation of the non linear Boltzmann collision
operator. The presented method is applicable to a general class of collision
kernels, among others including Maxwell molecules, hard and variable hard
spheres molecules. Although faster methods are available, we obtain high
accuracy even for very low expansion orders.

Numerical Analysis