the Coulomb phase shift was recovered after a renormalization procedure. We can conclude that the integral relations used in Eq. (48) produce the same effect as the renormalization procedure. In order to study applications of the integral relations using the HA method for a three- nucleon system, we give a brief introduction to the method following Refs. [22, 23] (for more details see Ref. [28]). In the HA method the three-body wave function of Eq. (35) is expanded as where Φ$_{µ}$( ρ, Ω) is a HA basis element and [ ρ, Ω] ≡ [ ρ, φ$_{i}$, ˆ X$_{i}$, ˆ Y$_{i}$ ] is the set of coordinates consisting of the hyperradius and of the five hyperspherical coordinates. The HA basis elements are the eigenfunctions of the hyperangular part of the Hamiltonian at fixed values of ρ : where G 2 is the grand-angular operator and V ( ρ, Ω) = ∑ i V ( X$_{i}$ ) is the potential energy. The eigenvalues, U$_{µ}$ ( ρ ), are the adiabatic potentials, that appear in the coupled set of differential equations with T$_{ρ}$ = ∂ $^{2}$/∂ρ2 + (5 /ρ ) ∂/∂ρ , N$_{A}$ the number of adiabatic channels included in the cal- culation, E the three-body energy, and from which the hyperradial functions w$_{µ}$ ( ρ ) are obtained. The coupling terms are defined as: P$_{µµ}$ ′ = < Φ$_{µ}$( ρ, Ω) | ∂/∂ρ | Φ$_{µ}$ ′ ( ρ, Ω) > and Q$_{µµ}$ ′ = < Φ$_{µ}$( ρ, Ω) | ∂ $^{2}$/∂ρ$^{2}$| Φ$_{µ}$ ′ ( ρ, Ω) > . In Ref. [22] the solution of the system of Eq. (51) has been studied for scattering states below the three-body breakup. In that study it emerged that the use of the integral relations helped to obtain a pattern of convergence for the phase-shift, in terms of the adiabatic channels, similar to the pattern obtained when the 18 [ − ℏ 2 2 $_{m}$T$_{ρ}$ + U$_{µ}$ ( ρ ) − ℏ 2 2 $_{m}$Q$_{µµ}$ ( ρ ) − E ] w$_{µ}$ ( ρ ) − ℏ 2 2 m N$_{A}$ ∑ µ ′ = µ [ Q$_{µµ}$ ′ ( ρ ) + P$_{µµ}$ ′ ( ρ )( 5 ρ + 2 d $_{dρ}$) w$_{µ}$ ′ ( ρ ) = 0 (51) [ ℏ 2 2 mρ $_{2}$G 2 + V ( ρ, Ω) ] Φ$_{µ}$( ρ, Ω) = U$_{µ}$ ( ρ )Φ$_{µ}$( ρ, Ω) , (50) Ψ = ∞ ∑ µ =1 w$_{µ}$ ( ρ )Φ$_{µ}$( ρ, Ω) , (49) V. INTEGRAL RELATIONS WITHIN THE HYPERSPHERICAL ADIABATIC METHOD