where we have made use of the fact that: With the above normalization, and assuming that Ψ is an exact solution of Eq.(1), it follows that Ψ satisfies the following integral relations: Explicitly they are where in the last integral we have used the property that ∇ $^{2}$(1 /r ) = − 4 πδ ( r ). In practical cases the solution of the Schr¨ odinger equation is obtained numerically. Then, tan δ is extracted from φ ( r ) analyzing its behavior outside the range of the potential. The equivalence between the extracted value and that obtained from the integral relations defines the accuracy of the numerical computation. A relative difference of the order of 10 − 7 of the two values is usually achieved using standard numerical techniques to solve the differential equation and to compute the two one-dimensional integrals. The short range character of the integral relations should be noticed. This means that the phase-shift is determined by the internal structure of the wave function. The second relation of Eq. (7) shows a dependence on the value of the wave function at the origin. It could be convenient to eliminate this explicit dependence since the numerical determination of φ (0) might be problematic, as we will show. To this end we introduce a regularized function ˜ G = f$_{reg}$G with the property that | ˜ G ( r = 0) | < ∞ and ˜ G = G outside the interaction region. A possible choice is where the regularization function f$_{reg}$ = (1 − e − $^{γr}$) has been introduced where γ is a non linear parameter which will be discussed below. Values satisfying γ > 1 /r$_{0}$ , with r$_{0}$ the 5 ˜ G = √ k 4 π cos( kr ) kr (1 − $_{e}$− $^{γr}$) , (8) − m ℏ 2 √$_{k}$ ∫ ∞ 0 dr sin( kr ) V ( r )[ rφ ( r )] = B m ℏ 2 √$_{k}$ ∞ 0 dr cos( kr ) V ( r )[ rφ ( r )] + φ (0) √$_{k}$ = A , (7) − m ℏ 2 < Ψ | H − E | F > = B m ℏ 2 < Ψ | H − E | G > = A tan δ = B A . (6) m ℏ 2 [ < F | H − E | G > − < G | H − E | F > ] = 1 . (5)