  The dynamical interplay between electron-electron interactions and
electron-phonon coupling is investigated within the Anderson-Holstein model, a
minimal model for open quantum systems that embody these effects. The influence
of phonons on spectral and transport properties is explored in equilibrium, for
non-equilibrium steady state and for transient dynamics after a quench. Both
the particle-hole symmetric and the more generic particle-hole asymmetric cases
are studied. The treatment is based on two complementary non-crossing
approximations, the first of which is constructed around the weak-coupling
limit and the second around the polaron limit. In general, the two methods
disagree in nontrivial ways, indicating that more reliable approaches to the
problem are needed. The frameworks used here can form the starting point for
numerically exact methods based on bold-line continuous-time quantum Monte
Carlo algorithms capable of treating open systems simultaneously coupled to
multiple fermionic and bosonic baths.

Strongly Correlated Electrons