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

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Abstract
In this paper and in the following, we study, in potential scattering, the existence and meaning of the solutions of the N/D equations in the equivalent formulation f/f. For S waves, considering only regular discontinuity μΔ(x), such that the resulting integral equation is of the Fredholm type, we study the corresponding Fredholm determinant 𝒟(μ). We remark that Marchenko formalism gives exactly the same resulting equation and then we have the possibility to interpret in terms of local potentials. We show the connection between the nth trace of the kernel of the resulting integral equation and the nth term of the potential reconstructed from the discontinuity. The connection between dispersion relation and the corresponding potential reconstructed from the discontinuity is given by the relation D(μ)=exp−12 ∫ 0∞ ∫ r∞V(t, μ) dt dr. In the present paper we limit our study to |μ| less than the smallest modulus root of 𝒟(μ) where a perturbation expansion of the solution exists and we show that V(r, μ) is regular at r = 0. On the other hand, for Yukawa-type potentials where the inverse Laplace transform is λC(α) the Fredholm determinant is exp (−∫m∞λC(α)/α2 dα) and cannot vanish such that the corresponding solutions of the resulting integral equation exist always.