  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.
