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

Feshbach projection-operator formalism to resonance scattering on Bargmann-type potentials