Description of the two-nucleon system on the basis of the Bargmann representation of the S matrix
For the effective-range function $k\cot \delta $, a pole approximation that involves a small number of parameters is derived on the basis of the Bargmann representation of the $S$ matrix. The parameters of this representation, which have a clear physical meaning, are related to the parameters of the Bargmann $S$ matrix by simple equations. By using a polynomial least-squares fit to the function $k\cot \delta $ at low energies, the triplet low-energy parameters of neutron-proton scattering are obtained for the latest experimental data of Arndt et al. on phase shifts. The results are $a_{t}=5.4030 $fm, $r_{t}=1.7494 $fm, and $v_{2}=0.163 $fm$^{3}$. With allowance for the values found for the low-energy scattering parameters and for the pole parameter, the pole approximation of the function $k\cot \delta $ provides an excellent description of the triplet phase shift for neutron-proton scattering over a wide energy range ($T_{\text{lab}}\lesssim 1000 $MeV), substantially improving the description at low energies as well. For the experimental phase shifts of Arndt et al., the triplet shape parameters $v_{n}$ of the effective-range expansion are obtained by using the pole approximation. The description of the phase shift by means of the effective-range expansion featuring values found for the low-energy scattering parameters proves to be fairly accurate over a broad energy region extending to energy values approximately equal to the energy at which this phase shift changes sign, this being indicative of a high accuracy and a considerable value of the effective-range expansion in describing experimental data on nucleon-nucleon scattering. The properties of the deuteron that were calculated by using various approximations of the effective-range function comply well with their experimental values.