variational estimate and the exact value reduces, for example at M = 40 it is about 10 − $^{6}$. In fact, as M increases, each eigenvector gives a better representation of the exact wave function in the internal region and the second order estimates, [tan δ$_{j}$ ] 2 nd approach the exact result. The study of the stability of the results in terms of the non linear parameter γ in the regularization function f$_{reg}$ is given in Fig. 2. In the upper panel, the second order [tan δ$_{1}$ ] 2 $^{nd}$, corresponding to the eigenvalue E$_{1}$ given in Table I, is shown as a function of γ , for M = 20 , 30 , 40. We observe a good stability for values of γ > 0 . $_{2 fm}$− 1 indicating that the regularization has to be done before ≈ 5 fm. In lower panel the corresponding values for I$_{γ}$ as a function of γ are shown. The stable values obtained for M = 20 , 30 , 40 when γ > 0 . $_{2 fm}$− 1 are I$_{γ}$ = − 8 . 6234 , − 8 . 4334 , − 8 . 3755 respectively. The exact values for φ (0) $_{/}$√$_{k}$$_{, obtained solving the Schr¨}$ odinger equation numerically at the three energies are: − 8 . 6188 , − 8 . 4338 , − 8 . 3755, respectively. We can observe that for M = 40 there is a com- plete agreement between I$_{γ}$ and φ (0) $_{/}$√$_{k}$$_{. Therefore}$ I$_{γ}$ can be considered to be an integral representation of φ (0) $_{/}$√$_{k}$$_{. This is an important point since such a value can be used to}$ normalize the variational wave function. In this example the integral relations derived from the KVP have been used to compute phase-shifts using bound state like wave functions. A different application of the integral relations regards the possibility of calculating the phase-shift of a process in which the two particles interact through a short range potential plus the Coulomb potential, imposing free asymptotic conditions to the wave function. As an example we use the same two body potential used in the previous analysis and add the Coulomb potential: For positive energies and l = 0, the wave function behaves asymptotically as with F$_{c}$ ( r ) , G$_{c}$ ( r ) the regular and irregular Coulomb functions, respectively. The phase- shift is tan δ$_{c}$ = B/A . The KVP remains valid when the long range Coulomb potential is 10 V ( r ) = − V$_{0}$ exp − ( r/r$_{0}$ ) 2 + e 2 r . (28) Ψ ( c $^{)}$( r → ∞ ) = AF$_{c}$ ( r ) + BG$_{c}$ ( r ) , (29)