  We study kink-antikink scattering in a one-parameter variant of the $\phi^4$
theory where the model parameter controls the static intersoliton force. We
interpolate between the limit of no static force (BPS limit) and the regime
where the static interaction is small (non-BPS). This allows us to study the
impact of the strength of the intersoliton static force on the soliton
dynamics.
  In particular, we analyze how the transition of a bound mode through the mass
threshold affects the soliton dynamics in a generic process, i.e., when a
static intersoliton force shows up. We show that the thin, precisely localized
spectral wall which forms in the limit of no static force, broadens in a
well-defined manner when a static force is included, giving rise to what we
will call a thick spectral wall. This phenomenon just requires that a discrete
mode crosses into the continuum at some intermediate stage of the dynamics and,
therefore, should be observable in many soliton-antisoliton collisions.
