  For fermions with degenerate single-particle energy levels, the usual
relation between the total number of particles and the chemical potential $\mu
$ is only satisfied for a specific number of particles, i.e. those leading to
closed shells. The treatment of an arbitrary number of fermions requires a
modification of the chemical potential, similar to the one proposed by
Landsberg for Bose-condensed systems. We study the implications of the required
modification for fermions in a potential, by calculating the ground state
energy, the free energy, the density, the partition function and the dynamic
two-point correlation function. It turns out that the modified relation between
the fugacity and the number of particles leads to the correct ground state
energy and density. But for other quantities like the entropy and the two-point
correlation functions, an additional correction is required and derived. These
calculations indicate that many-body perturbation theories based on $H-\mu N$
with Lagrange multiplier $\mu $, are not applicable in unmodified form for a
fixed number of fermions at low temperature.
