In this dissertation, we analyze both many- and few-body systems under external confinement with tunable interactions. First, we develop a density-renormalization ap- proach for describing two-component fermionic systems with short-range interactions. This renormalized zero-range interaction eliminates the instabilities produced by a bare Fermi pseudopotential and provides a simple description of the interactions from the weakly interacting BCS region up to unitarity.
In the second part of the thesis, we focus on few-body systems in the BCS-BEC crossover. To obtain the solutions, we implement two different numerical techniques: a correlated-Gaussian-basis-set expansion and a fixed-node diffusion Monte Carlo tech- nique. We also develop an innovative numerical technique for obtaining solutions to the four-body problem in the hyperspherical representation.
Our solutions provide an accurate description of few-body trapped systems. The analysis of two-, three-, and four-body systems, for instance, provides a few-body per- spective on the BCS-BEC crossover problem. The analysis of the spectrum of such systems allows us to visualize important pathways for molecule formation. We then use the four-body solutions to extract key properties of the system such as the dimer-dimer scattering length and the effective range.
}, year = {2008}, publisher = {University of Colorado Boulder}, address = {Boulder}, }