Fermions in a Multi-Soliton Background in 1+1 Dimensions

We consider the fermionic bound states associated with a soliton-antisoliton pair in 1+1 dimensions which have zero energy when the solitons are infinitely far apart. We calculate the energies of these states when the solitons are separated by a finite distance. The energies decay exponentially with the distance between the soliton and antisoliton. When the fermion mass is much larger than the boson mass, the energy simplifies substantially. These energies may be interpreted as a contribution to the effective potential between the soliton and antisoliton. The character of this contribution depends upon which fermionic states are occupied. Performing the analogous calculation for the simplest (3+1)-dimensional soliton system, we find no fermionic energy shift.