  Consequences of a non-trivial scalar field background for an effective 4D
theory were studied in the context of one compact extra dimension. The periodic
background that appears within the (1+4)-dimensional $\phi^4$ theory was found
and the excitations above the background (and their spectrum) were determined
analytically. It was shown that the presence of the non-trivial solution leads
to the existence of a minimal size of the extra dimension that is determined by
the mass parameter of the scalar potential. It was proved that imposing
orbifold antisymmetry boundary conditions allows us to eliminate a negative
mass squared Kaluza-Klein ground-state mode that otherwise would cause an
instability of the system. The localization of fermionic modes in the presence
of the non-trivial background was discussed in great detail varying the size of
the extra dimension and the strength of the Yukawa coupling. A simple exact
solution for the zero-mode fermionic states was found and the solution for
non-zero modes in terms of trigonometric series was constructed. The fermionic
mass spectrum, which reveals a very interesting structure, was found
numerically. It was shown that the natural size of the extra dimension is twice
as large as the period of the scalar background solution.
