We compare the BCS approximation to exact diagonalization for the 1 and 2 dimensional attractive hubbard model. While it has been reported before that the BCS accurately reproduces the energy at even filling, in one dimension, the current work looks at correlators as well and finds the approximation somewhat lacking. On the other hand, it appears to be somewhat better in 2 dimensions and is a very good approximation to the exact diagonalization of a reduced hamiltonian which may have physical interest in its own right.
