content_image/332812.jpg content_image/332811.jpg content_image/332810.jpg content_image/332809.jpg content_image/332808.jpg content_image/332807.jpg
The variation of the functional with respect to the linear parameters a$_{n}$ and tan δ leads to the following equations 

<img src='content_image/332812.jpg'>

To obtain the last equation, the normalization relation of Eq. (9) has been used. From these two equations, Ψ$_{c}$ and the first order estimate of the phase shift (tan δ ) 1 st can be determined. It should be noted that the first equation implies < Ψ$_{c}$ | H − E | Ψ$_{t}$ > = 0. Furthermore, from the general relation for A in Eq. (4), and using the second equation in Eq. (16), the following integral relation results 

<img src='content_image/332811.jpg'>

Replacing the two relations of Eq.(16) into the functional of Eq.(14), a second order estimate of the phase shift is obtained 

<img src='content_image/332810.jpg'>

Multiplying Eq. (18) by A one gets 

<img src='content_image/332809.jpg'>

On the other hand, a first order estimate for the coefficient B can be obtained from the general relation in Eq. (4), i.e., 

<img src='content_image/332808.jpg'>

Therefore, replacing Eq.(20) in Eq.(19), a second order integral relation for B is obtained. The above results can be summarized as follow 

<img src='content_image/332807.jpg'>

These equations extend the validity of the integral relations, given in Eq.(6) for the exact wave functions, to trial wave functions. To be noticed that F, ˜ G are solutions of the Schr¨ odinger equation in the asymptotic region, therefore ( H − E ) F → 0 and ( H − E ) ˜ G → 0 as the distance between the particles increases. As a consequence the decomposition of 

