  We present an exact diagonalization study of the self-energy of the
two-dimensional Hubbard model. To increase the range of available cluster sizes
we use a corrected t-J model to compute approximate Greens functions for the
Hubbard model. This allows to obtain spectra for clusters with 18 and 20 sites.
The self-energy has several `bands' of poles with strong dispersion and
extended incoherent continua with k-dependent intensity. We fit the self-energy
by a minimal model and use this to extrapolate the cluster results to the
infinite lattice. The resulting Fermi surface shows a transition from hole
pockets in the underdoped regime to a large Fermi surface in the overdoped
regime. We demonstrate that hole pockets can be completely consistent with the
Luttinger theorem. Introduction of next-nearest neighbor hopping changes the
self-energy stronlgy and the spectral function with nonvanishing
next-nearest-neighbor hopping in the underdoped region is in good agreement
with angle resolved photoelectron spectroscopy.
