Spectral analysis of Dirac operators for fermion scattering on topological solitons in the nonlinear $O(3)$ $\sigma$-model

We investigate the existence of discrete positive or negative energy ground states of the Dirac operator $H$ which describe the fermion scattering on topological solitons in the nonlinear $O(3)$ $\sigma$-model. Additionally, we provide a sufficient condition to ensure that the positive and negative energies of the Dirac operator $H$ are non-zero.