Small-data scattering for nonlinear waves with potential and initial data of critical decay

We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp global existence results for the Cauchy problem with small initial data; those require the data to decay at a rate $k\geq k_c$, where $k_c$ is a critical decay rate that depends on the order of the nonlinearity. However, scattering results have appeared only for the supercritical case $k>k_c$. In this paper, we extend the latter results to the critical case and we also allow the presence of a short-range potential.