On Lieb-Robinson bounds for a class of continuum fermions

We consider the quantum dynamics of a many-fermion system in $\mathbb R^d$ with an ultraviolet regularized pair interaction as previously studied in [M. Gebert, B. Nachtergaele, J. Reschke, and R. Sims, Ann. Henri Poincar\'e 21.11 (2020)]. We provide a Lieb-Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb-Robinson bound on $L^2$-overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb-Robinson bounds.