  We present closed analytical expressions for the particle and kinetic energy
spatial densities at finite temperatures for a system of noninteracting
fermions (bosons) trapped in a d-dimensional harmonic oscillator potential. For
d=2 and 3, exact expressions for the N-particle densities are used to calculate
perturbatively the temperature dependence of the splittings of the energy
levels in a given shell due to a very weak interparticle interaction in a
dilute Fermi gas. In two dimensions, we obtain analytically the surprising
result that the |l|-degeneracy in a harmonic oscillator shell is not lifted in
the lowest order even when the exact, rather than the Thomas-Fermi expression
for the particle density is used. We also demonstrate rigorously (in two
dimensions) the reduction of the exact zero-temperature fermionic expressions
to the Thomas-Fermi form in the large-N limit.
