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

TL;DR

This paper applies the Feshbach projection-operator formalism to resonance scattering in Bargmann-type potentials, analyzing the nature of S-matrix poles and their relation to physical resonances and cut-off effects.

Contribution

It demonstrates that the Feshbach formalism accurately identifies physical and spurious S-matrix poles in generalized Bargmann-type potentials with resonances.

Findings

Feshbach formalism correctly distinguishes physical and spurious poles.

Numerical analysis confirms pole correspondence with physical resonances.

Study extends Bargmann potentials to include resonance states.

Abstract

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.