Fermions on one or fewer kinks

We find the full spectrum of fermion bound states on a Z2 kink. In addition to the zero mode, there are int[2mf/ms] bound states, where mf is the fermion and ms the scalar mass. We also study fermion modes on the background of a well-separated kink-antikink pair. Using a variational argument, we prove that there is at least one bound state in this background, and that the energy of this bound state goes to zero with increasing kink-antikink separation, 2L, and faster than e-a2L where a=min(ms,2mf). By numerical evaluation, we find some of the low lying bound states explicitly.