content_image/330701.jpg content_image/330700.jpg content_image/330699.jpg content_image/330698.jpg
is decomposed in three Faddeev-like amplitudes 

<img src='content_image/330701.jpg'>

where we have introduced the Jacobi coordinates: X i = ( r j − r $_{k}$) $_{/}$√$_{2 and}$ Y i = ( r j + r k − 2 r $_{i}$) $_{/}$√$_{6 (the generic vector}$ r k indicates the position of nucleon k ). Each amplitude having quantum numbers J, J$_{z}$, T, T$_{z}$ is expanded in angular-spin-isospin channels (called α -channels) as 

<img src='content_image/330700.jpg'>

and the radial amplitudes are expanded in terms of the PHH basis 

<img src='content_image/330699.jpg'>

where we have introduced the hyperspherical variables, the hyperradius ρ and the hyperangle φ$_{i}$ , defined by the relations X$_{i}$ = ρ cos φ$_{i}$, Y$_{i}$ = ρ sin φ$_{i}$ , and $^{(2)}$P l$_{α}$,L$_{α}$ n ( φ$_{i}$ ) is a hyperspherical polynomial. The summation is given in terms of the grand angular quantum number K = 2 n + l$_{α}$ + L$_{α}$ . The correlation functions f$_{α}$ ( r ) are introduced to accelerate the rate of convergence of the expansion. They take into account those configurations in which two nucleons are close to each other. A very convenient choice is to derive the correlation functions from a Schr¨ odinger like equation governed by the two-body potential corresponding to the specific α -channel [25]. 

In the following we consider a three-nucleon system in either the J = 1 / 2 + or J = 3 / 2 + states with total isospin T = 1 / 2. The central character of the MT I-III interaction decouples those channels with different values of Λ$_{α}$. Moreover, as the interaction acts only in the s - wave, we have l$_{α}$ = 0. This condition limits the number of channels of the (Λ$_{α}$ = 0) J = 1 / 2 + state to two channels, corresponding to s jk α = 0 , 1, and to one channel, corresponding to s jk α = 1, in the case of the (Λ$_{α}$ = 0) J = 3 / 2 + state. Finally, following Ref. [26], the hyperradial functions are expanded in terms of Laguerre polynomials 

<img src='content_image/330698.jpg'>

with z = βρ , and β a non linear parameter. We can define a complete antisymmetric three- nucleon state | α, K, m > , in terms of which the wave function Ψ$_{n}$ for the n -th state results 

