Two-Channel Effective Range Expansion and Bound-State Properties

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 $${\epsilon}$$ and are expressed in terms of matrix elements c ij = (K −1) ij where K is the reaction K matrix. Quantities $${g_{ij}(k)=k^{l_i+l_j+1}c_{ij}(k)}$$ 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.