Fermions on one or fewer kinks

We find the full spectrum of fermion bound states on a ${Z}_{2}$ kink. In addition to the zero mode, there are $\mathrm{int}[2{m}_{f}/{m}_{s}]$ bound states, where ${m}_{f}$ is the fermion and ${m}_{s}$ 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 ${e}^{\ensuremath{-}a2L}$ where $a=\mathrm{min}({m}_{s},2{m}_{f})$. By numerical evaluation, we find some of the low lying bound states explicitly.