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Spectral intensity distribution of trapped fermions

To calculate static response properties of a many body system, local density approximation (LDA) can be safely applied. But applicability of LDA is limited for the case of dynamical response functions since dynamics of the system needs to be considered as well. To examine this in the context of cold atoms, we consider a system of non-interacting spin-$1/2$ fermions confined by a harmonic trapping potential. We have calculated a very important response function, the spectral intensity distribution function (SIDF), both exactly and using LDA at zero temperature and compared with each other for different dimensions, trap frequencies and momenta. The behavior of the SIDF at a particular momentum can be explained by noting the behavior of the density of states (DOS) of the free system (without trap) in that particular dimension. The agreement between exact and LDA SIDFs becomes better with increase in dimensions and number of particles.