  We study the edge behavior of inhomogeneous one-dimensional quantum systems,
such as Lieb-Liniger models in traps or spin chains in spatially varying
magnetic fields. For free systems these fall into several universality classes,
the most generic one being governed by the Tracy-Widom distribution. We
investigate in this paper the effect of interactions. Using semiclassical
arguments, we show that since the density vanishes to leading order, the strong
interactions in the bulk are renormalized to zero at the edge, which simply
explains the survival of Tracy-Widom scaling in general. For integrable
systems, it is possible to push this argument further, and determine exactly
the remaining length scale which controls the variance of the edge
distribution. This analytical prediction is checked numerically, with excellent
agreement. We also study numerically the edge scaling at fronts generated by
quantum quenches, which provide new universality classes awaiting theoretical
explanation.
