The Klein-Gordon equation in anti-de Sitter spacetime - Seminario ...

284 K. Yagdjian and A. Galstian=∫ t0∫ e t −10]db[14e −t/2 e b/2 (2+b)+16 1 (1+4M2 )be −3t/2 e b/2 (e b − e t )[]× ϕ(x+e t − e b )+ϕ(x−e t + e b )[ϕ(x+z)+ϕ(x−z)][14 e−t/2 (2+ln(e t − z))]−16 1 (1+4M2 )e −3t/2 zln(e t 1− z) √e t − z dz.Thenu(x,t) =∫ e t −10[ϕ(x+z)+ϕ(x−z)][14 e−t/2 (2+ln(e t − z))]−16 1 (1+4M2 )e −3t/2 zln(e t 1− z) √e t − z dz∫ t ∫ e t −e b [ ]+ db dz ϕ(x+z)+ϕ(x−z)0 0[ ( ∂ 2 ]× be∂z) 2b E(z,t;0,b)−M 2 bE(z,t;0,b) .In the last integral we change the order of integration,u(x,t) =∫ e t −10[ϕ(x+z)+ϕ(x−z)][14 e−t/2 (2+ln(e t − z))]− 1 16 (1+4M2 )e −3t/2 zln(e t 1− z) √e t − z dz∫ e t −1 [ ]∫ ln(e t −z)+ dz ϕ(x+z)+ϕ(x−z) dbb00[ ( ∂ 2 ]× e∂z) 2b E(z,t;0,b)−M 2 E(z,t;0,b) ,and obtain (27), where K 1 (z,t) is defined by (29). The corollary is proven.The next lemma completes the proof of Theorem 4 in the case of ϕ 0 = 0.LEMMA 1. The kernel K 1 (z,t) defined by (29) coincides with one given in Theorem4.Proof. Due to Lemma 1.2 [16], (19), and by integration by parts, we have[]∫ ln(e t −z)0be 2b ( ∂∂z) 2E(z,t;0,b)−M 2 E(z,t;0,b)[ ∂]= ln(e t − z)∂b E(z,t;0,b) − b=ln(e t −z) E(z,t;0,ln(et − z))+E(z,t;0,0).db