is decomposed in three Faddeev-like amplitudes 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 and the radial amplitudes are expanded in terms of the PHH basis 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 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 13 u α $_{K}$( ρ ) = $^{M}$− 1 ∑ m =0 A α K,m L (5) m ( z ) exp ( − z/ 2) (38) Φ$_{α}$( X$_{i}$, Y$_{i}$ ) = ρ l$_{α}$ + $^{L$_{α}$}$f$_{α}$ $^{$_{(}$√$_{2}$$_{X}$$_{i}$$_{)}$}$∑ K u α $_{K}$( ρ ) $^{(2)}$P l$_{α}$,L$_{α}$ K ( φ$_{i}$ ) (37) ψ ( X $_{i}$, Y $_{i}$) = ∑ α Φ$_{α}$( X$_{i}$, Y$_{i}$ ) Y $_{α}$( jk, i ) Y $_{α}$( jk, i ) = { [ Y$_{l}$$_{α}$ ( ˆ X$_{i}$ ) ⊗ Y$_{L}$$_{α}$ ( ˆ Y$_{i}$ )]$_{Λ}$$_{α}$[ s jk α ⊗ 1 $_{2}$]$_{S}$$_{α}$ } $_{JJ$_{z}$}$[ t jk α ⊗ 1 $_{2}$]$_{T T}$$_{z}$ (36) Ψ = ψ ( X $_{i}$, Y $_{i}$) + ψ ( X $_{j}$, Y $_{j}$) + ψ ( X $_{k}$, Y $_{k}$) , (35)