36 Furthermore, eq. (A5b) splits into three independent conditions: [1] S. Weinberg, Phys. Rev. D13 , 974 (1976). [2] L. Susskind, Phys. Rev. D20 , 2619 (1979). [3] S. Weinberg, Phys. Rev. D19 , 1277 (1979). [4] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B429 , 263 (1998), [hep-ph/9803315]. [5] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 , 3370 (1999), [hep-ph/9905221]. [6] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 , 4690 (1999), [hep-th/9906064]. [7] J. M. Maldacena, Adv. Theor. Math. Phys. 2 , 231 (1998), [hep-th/9711200]. [8] C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Phys. Rev. D69 , 055006 (2004), [hep-ph/0305237]. [9] K. Agashe, A. Delgado, M. J. May and R. Sundrum, JHEP 08 , 050 (2003), [hep-ph/0308036]. Rn ∣$^{0}$ b N n δ$_{mn}$ = − m 2 $_{N n}$δ$_{mn}$ + m 2 N n ˜ g 2 ( ˜ f 3 Lm ˜ f 3 Ln + ˜ f 3 Rm ˜ f 3 Rn ) ∣ πR + ˜ v 2 4 b ( πR ) ( ˜ f 3 Lm ˜ f 3 Ln + ˜ f 3 Lm ˜ f 3 Ln − 2 ˜ f 3 Lm ˜ f 3 Ln ) ∣ πR + b ( πR ) ( ˜ f 3 $_{Lm}$∂$_{y}$ ˜ f 3 Ln + ˜ f 3 Rm ˜ f 3 Rn ) ∣ ∣$^{0}$ $_{πR}$, (A16) f a obey the BCs (17), (18), (24), (25), (30) Ln ∣$^{0}$ b R $_{n}$δ$_{mn}$ = − m 2 $_{Rn}$δ$_{mn}$ + ( m 2 Rn ˜ g 2 + ˜ v 2 4 b ( πR ) ) ˜ f 1 , 2 Rm ˜ f 1 , 2 Rn ∣ πR + b ( πR ) ( ˜ f 1 , 2 $_{Rm}$∂$_{y}$ ˜ f 1 , 2 Rn ) ∣ $_{πR}$, (A15) m 2 b L $_{n}$δ$_{mn}$ = − m 2 $_{Ln}$δ$_{mn}$ + ( m 2 Ln ˜ g 2 + ˜ v 2 4 b ( πR ) ) ˜ f 1 , 2 Lm ˜ f 1 , 2 Ln ∣ πR + b ( πR ) ( ˜ f 1 , 2 $_{Lm}$∂$_{y}$ ˜ f 1 , 2 Ln ) ∣ $_{πR}$, (A14) m 2 ˜ v 2 which are identically satisfied as soon as the f L,Rn obey the BCs (17), (18), (24), (25), (30) and (31). Notice that eq. (15), (23), (29) together with the above mentioned BCs guarantee the orthogonality of the wave-functions that we assumed in eq. (A13), so we have a self- consistent solution of eqs. (A5a) and (A5b). To complete the diagonalization and finally get an expanded bilinear Lagrangian, we just need to solve the last two equations in the set (A5). This can be obtained by imposing the conditions (19), (26) and (32) respectively on the scalar profiles of the three sectors.