  The projection-operator formalism of Feshbach is applied to resonance
scattering in a single-channel case. The method is based on the division of the
full function space into two segments, internal (localized) and external
(infinitely extended). The spectroscopic information on the resonances is
obtained from the non-Hermitian effective Hamilton operator $H_{\rm eff}$
appearing in the internal part due to the coupling to the external part. As
well known, additional so-called cut-off poles of the $S$-matrix appear,
generally, due to the truncation of the potential. We study the question of
spurious $S$ matrix poles in the framework of the Feshbach formalism. The
numerical analysis is performed for exactly solvable potentials with a finite
number of resonance states. These potentials represent a generalization of
Bargmann-type potentials to accept resonance states. Our calculations
demonstrate that the poles of the $S$ matrix obtained by using the Feshbach
projection-operator formalism coincide with both the complex energies of the
physical resonances and the cut-off poles of the $S$-matrix.
