  The first- and second-order correlation functions of trapped, interacting
Bose-Einstein condensates are investigated numerically on a many-body level
from first principles. Correlations in real space and momentum space are
treated. The coherence properties are analyzed. The results are obtained by
solving the many-body Schr\"odinger equation. It is shown in an example how
many-body effects can be induced by the trap geometry. A generic fragmentation
scenario of a condensate is considered. The correlation functions are discussed
along a pathway from a single condensate to a fragmented condensate. It is
shown that strong correlations can arise from the geometry of the trap, even at
weak interaction strengths. The natural orbitals and natural geminals of the
system are obtained and discussed. It is shown how the fragmentation of the
condensate can be understood in terms of its natural geminals. The many-body
results are compared to those of mean-field theory. The best solution within
mean-field theory is obtained. The limits in which mean-field theories are
valid are determined. In these limits the behavior of the correlation functions
is explained within an analytical model.
