  A recent work on the resummation of fermionic in-medium ladder diagrams to
all orders is extended by calculating the complex single-particle potential
$U(p,k_f)+ i\,W(p,k_f)$ for momenta $p<k_f$ as well as $p>k_f$. The on-shell
single-particle potential is constructed by means of a complex-valued in-medium
loop that includes corrections from a test-particle of momentum $\vec p$ added
to the filled Fermi sea. The single-particle potential $U(k_f,k_f)$ at the
Fermi surface as obtained from the resummation of the combined particle and
hole ladder diagrams is shown to satisfy the Hugenholtz-Van-Hove theorem. The
perturbative contributions at various orders $a^n$ in the scattering length are
deduced and checked against the known analytical results at order $a^1$ and
$a^2$. The limit $a\to\infty$ is studied as a special case and a strong
momentum dependence of the real (and imaginary) single-particle potential is
found. This indicates an instability against a phase transition to a state with
an empty shell inside the Fermi sphere such that the density gets reduced by
about 5%. For comparison, the same analysis is performed for the resummed
particle-particle ladder diagrams alone. In this truncation an instability for
hole-excitations near the Fermi surface is found at strong coupling. For the
set of particle-hole ring diagrams the single-particle potential is calculated
as well. Furthermore, the resummation of in-medium ladder diagrams to all
orders is studied for a two-dimensional Fermi gas with a short-range two-body
contact-interaction.
