content_image/336912.jpg content_image/336911.jpg content_image/336910.jpg content_image/336909.jpg
considered and its form in terms of the integral relations results to be: 

<img src='content_image/336912.jpg'>

with ˜ G$_{c}$ = f$_{reg}$G$_{c}$ and Ψ ( c ) t a trial wave function behaving asymptotically as Ψ ( c $^{)}$. Since ( H − E ) | F$_{c}$ > and ( H − E ) | ˜ G$_{c}$ > go to zero outside the range of the short range potential, the integrals in Eq. (30) are negligible outside that region. Therefore, for the computation of the phase-shift it is enough to require that Ψ ( c ) t satisfies ( H − E )Ψ ( c ) t = 0, inside that region. To exploit this fact, we introduce the following screened potential: 

For specific values of n and r$_{sc}$ it has the property of being extremely close to the potential V ( r ) of Eq. (28) for r < r$_{0}$ , with r$_{0}$ the range of the short range potential. The screening factor e − ( r/r$_{sc}$ ) n cuts the Coulomb potential for r > r$_{sc}$ . Using the potential V$_{sc}$ to describe a scattering process, the wave function behaves asymptotically as 

<img src='content_image/336911.jpg'>

<img src='content_image/336910.jpg'>

where F,G are given by Eq. (27), since V$_{sc}$ is a short range potential. It should be noted that the screened phase-shift tan δ$_{n,r}$$_{sc}$ does not equal tan δ$_{c}$ for any finite value of n and r$_{sc}$ . Solving the Schr¨ odinger equation for V$_{sc}$ , it is possible to obtain the wave function Ψ$_{n,r}$$_{sc}$ for different values of n and r$_{sc}$ . This wave function can be considered to be a trial wave function for the problem in which the Coulomb potential is unscreened. Accordingly it can be used as input in Eq. (30) to obtain a second order estimate of the Coulomb phase-shift, 

<img src='content_image/336909.jpg'>

where the unscreened Coulomb potential is included in H . This estimate depends on n and r$_{sc}$ as the wave function does. In Fig. 3 the second order estimate [tan δ$_{c}$ ] 2 nd is shown as a function of r$_{sc}$ for different values of n . The straight line is the exact value of tan δ$_{c}$ obtained 

