  We consider a fermion chirally coupled to a prescribed pseudoscalar field in
the form of the soliton of the sine-Gordon model and calculate and investigate
the Casimir energy and all of the relevant quantities for each parity channel,
separately. We present and use a simple prescription to construct the
simultaneous eigenstates of the Hamiltonian and parity in the continua from the
scattering states. We also use a prescription we had introduced earlier to
calculate unique expressions for the phase shifts and check their consistency
with both the weak and strong forms of the Levinson theorem. In the graphs of
the total and parity decomposed Casimir energies as a function of the
parameters of the pseudoscalar field distinctive deformations appear whenever a
fermionic bound state energy level with definite parity crosses the line of
zero energy. However, the latter graphs reveal some properties of the system
which cannot be seen from the graph of the total Casimir energy. Finally we
consider a system consisting of a valence fermion in the ground state and find
that the most energetically favorable configuration is the one with a soliton
of winding number one, and this conclusion does not hold for each parity,
separately.
