  Spatial and momentum correlations are important in the analysis of the
quantum states and different phases of trapped ultracold atom systems as a
function of the strength of interatomic interactions. Identification and
understanding of spin-resolved patterns exhibited in two-point correlations,
accessible directly by experiments, are key for uncovering the symmetry and
structure of the many-body wave functions of the trapped system. Using the
configuration interaction method for exact diagonalization of the many-body
Hamiltonian of $N=2-4$ fermionic atoms trapped in single, double, triple, and
quadruple wells, we analyze both two-point momentum and space correlations, as
well as associated noise distributions, for a broad range of interparticle
contact repulsion strengths and interwell separations, unveiling
characteristics allowing insights into the transition, via an intermediate
phase, from the non-interacting Bose-Einstein condensate to the weakly
interacting quasi-Bose-Einstein regime, and from the latter to the
strong-repulsion Tonks-Girardeau (TG) one. The ab-initio numerical predictions
are shown to agree well with the results of a constructed analytical model
employing localized displaced Gaussian functions to represent the $N$ fermions.
The two-point momentum correlations are found to exhibit damped oscillatory
diffraction behavior. This diffraction behavior develops fully for atoms
trapped in a single well with strong interatomic repulsion in the TG regime, or
for atoms in well-separated multi-well traps. Additionally, the two-body
momentum correlation and noise distributions are found to exhibit
"shortsightedness", with the main contribution coming from nearest-neighboring
particles.
