Geodesic
A unitarity criterion for the partial $S$-matrix of resonance scattering
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 1, pp. 123-133
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We consider the influence of $N$ long-lived states characterized by resonance energies $E_i$ and widths $\Gamma_i(E)$ on the elastic scattering process and obtain an expression for the partial $S$-matrix $S_l(E)$ in the form of a sum over the resonance levels $($poles$)$ at which the residues have the form $\Gamma_i\prod_{\substack{k=1,\\k\ne i}}^N\gamma_{ik}$, where $\gamma_{ik}={(z_i-z_k^*)\imath/2(z_i-z_k)}$ and $z_i=E_i-\imath\Gamma_i/2$. We show that a necessary condition for the unitarity of the partial $S$-matrix in the presence of $N$ resonance levels can be written as $\sum_{i=1}^N \Gamma_i(E)\prod_{\substack{k=1,\\k\ne i}}^N\gamma_{ik}= \sum_{i=1}^N\Gamma_i(E)$.
Mots-clés : partial $S$-matrix, pole
Keywords: resonance level, unitarity, elastic scattering, necessary criterion for unitarity. 
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     author = {V. A. Khangulyan},
     title = {A~unitarity criterion for the~partial $S$-matrix of resonance scattering},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {123--133},
     year = {2019},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_1_a6/}
}
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V. A. Khangulyan. A unitarity criterion for the partial $S$-matrix of resonance scattering. Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 1, pp. 123-133. http://geodesic.mathdoc.fr/item/TMF_2019_199_1_a6/

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