  While the well-established $GW$ approximation corresponds to a resummation of
the direct ring diagrams and is particularly well suited for weakly-correlated
systems, the $T$-matrix approximation does sum ladder diagrams up to infinity
and is supposedly more appropriate in the presence of strong correlation. Here,
we derive and implement, for the first time, the static and dynamic
Bethe-Salpeter equations when one considers $T$-matrix quasiparticle energies
as well as a $T$-matrix-based kernel. The performance of the static scheme and
its perturbative dynamical correction are assessed by computing the neutral
excited states of molecular systems. Comparison with more conventional schemes
as well as other wave function methods are also reported. Our results suggest
that the $T$-matrix-based formalism performs best in few-electron systems where
the electron density remains low.
