Two-Channel Effective Range Expansion and Bound-State Properties

摘要

The S matrix and the f matrix amplitudes are considered for the case of two coupled elastic scattering channels, which differ in values of orbital angular momenta. Matrix elements of S and f matrices are parametrized in terms of scattering phases δ _i (i = 1, 2) and a mixing parameter e{epsilon} and are expressed in terms of matrix elements c _ij = (K −1) _ij where K is the reaction K matrix. Quantities g_ij(k)=kl_i+l_j+1c_ij(k){g_{ij}(k)=k^{l_i+l_j+1}c_{ij}(k)} are expanded in powers of k 2, k being the relative momentum of colliding particles B and C. Then functions g _ij (k) and c _ij (k) are continued analytically to the pole of amplitudes f _ij corresponding to the bound state A of colliding particles. This procedure allows to get the position of the pole as well as the residues of amplitudes f _ij at that pole which are related directly to vertex constants and asymptotic normalization coefficients corresponding to the vertex A → B + C.