  The intrinsically relativistic problem of a fermion subject to a pseudoscalar
screened Coulomb plus a uniform background potential in two-dimensional
space-time is mapped into a Sturm-Liouville. This mapping gives rise to an
effective Morse-like potential and exact bounded solutions are found. It is
shown that the uniform background potential determinates the number of
bound-state solutions. The behaviour of the eigenenergies as well as of the
upper and lower components of the Dirac spinor corresponding to bounded
solutions is discussed in detail and some unusual results are revealed. An
apparent paradox concerning the uncertainty principle is solved by recurring to
the concepts of effective mass and effective Compton wavelength.
