Connection between Marchenko Formalism and N/D Equations: Regular Interactions. II

In a previous paper we have shown, for S‐waves, that the resulting integral equations of the f/f equations (equivalent to the N/D approach) can be obtained from the Marchenko formalism. The potential V(μ, r) reconstructed from the discontinuity μΔ(x) is $\text{−2(d/dr)[(d}\mathcal{D}\mathrm{/dr)/}\mathcal{D}]$ , where $\mathcal{D}\mathrm{(\mu ,r)}$ is the Fredholm denominator of the Jost solution and $\mathcal{D}\text{(μ,0)}$ that of the resulting integral equation. For ``regular discontinuities'' we find different classes of V(μ, r). First, if $\mathcal{D}\text{(μ,0)=0}$ , then V(μ, r) is not ``regular at the origin'' [in general, we find that V(μ, r) becomes marginally singular: repulsive and singular like r⁻²]. Secondly, if $\mathcal{D}\text{(μ,0)≠0}$ , then V(μ, r) is ``regular'' at the origin and we obtain the following: (i) If |μ| is less than the smallest modulus root of $\mathcal{D}\text{(μ,0)}$ , then V(μ, r) has no poles for r ≥ 0. This range of |μ|‐values where the iteration series of the resulting integral equations converge is limited by the smallest |μ| value where a real or complex ghost can appear or where a bound state can appear at zero energy. (ii) For |μ| larger than this smallest modulus root but μ inside the interval given by the first positive and negative roots, V(μ, r) has no second‐order poles for r ≥ 0. These results (i) and (ii) are obtained with the restriction that in the considered interval there do not exist (μ, r)‐values such that $\mathcal{D}\text{(±μ,r)=0}$ , and from our study we cannot conclude that this is always true. (iii) For μ outside the above interval, V(μ, r) has poles of the second order for r > 0, the ``bound states'' being, in general, real or complex ghosts, or ``bound states'' corresponding to badly behaved potentials. We find also that the Jost solutions for energy equal to zero are $\mathcal{D}\mathrm{(-\mu ,r)/}\mathcal{D}\mathrm{(\mu ,r)}$ . This gives the connection between ghosts and, in general, possible bound states appearing at zero energy; this gives also the relations between poles of V(μ, r) corresponding to opposite μ values. These results for the Jost function correspond to a normalization at infinity, so we have considered the problem of subtractions with normalization at an arbitrary point. Then the new Fredholm determinant is the product of the old one by the value of the Jost function at the subtracted point. It follows that, if the first μ‐greater‐than‐zero and the first μ‐lessthan‐zero roots of $\mathcal{D}\mathrm{(\mu ,r)}$ are not opposite (r ≥ 0), the |μ| interval of convergence of the iteration series is enlarged for the subtracted equation.