Stable branches of a solution for a fermion on domain wall

The case when a fermion occupies an excited nonzero frequency level in the field of domain wall is discussed. It is demonstrated that a solution exists for the coupling constant in the limited interval 1 < g < g max ≈ 1.65. It is shown that indeed there are different branches of stable solution for g in this interval. The first one corresponds to a fermion located on the domain wall (\(1 < g < \sqrt[4]{{2\pi }}\)). The second branch, which belongs to the interval \(\sqrt[4]{{2\pi }} \leqslant g \leqslant g_{\max }\), describes a polarized fermion off the domain wall. The third branch with 1 < g < g max describes an excited antifermion in the field of the domain wall.