By (3.8), we calculate Then we have where with the conventions z · z ′ = N ∑ i =1 z$_{i}$z ′ i for all z = ( z$_{1}$, ..., z$_{N}$ ) ∈ C $^{N}$, z ′ = ( z ′ $_{1}$, ..., z ′ N ) ∈ C $^{N}$. Then we have where Re ( z ) is the real part of z. Now, we calculate the last term in (3.12) by using (3.10) and (3.11), we obtain P$_{s}$z = e $^{sϕ}$L$_{0}$ u = i∂$_{t}$z − is∂$_{t}$ϕz + Δ$_{y}$ z − Δ$_{x}$ z − 2 s ( ∇ $_{y}$ϕ · ∇ $_{y}$z − ∇ $_{x}$ϕ · ∇ $_{x}$z ) − s (Δ$_{y}$ ϕ − Δ$_{x}$ ϕ ) z + s 2 ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) z. P$_{s}$z + is∂$_{t}$ϕz = P + s z + P − s z, (3.9) P + s z = i∂$_{t}$z + Δ$_{y}$ z − Δ$_{x}$ z + s 2 ( |∇ $_{y}$ϕ | 2 − |∇ $_{x}$ϕ | 2 ) z, (3.10) P − s z = − 2 s ( ∇ $_{y}$ϕ · ∇ $_{y}$z − ∇ $_{x}$ϕ · ∇ $_{x}$z ) − s (Δ$_{y}$ ϕ − Δ$_{x}$ ϕ ) z, (3.11) ‖ P$_{s}$z + is∂$_{t}$ϕz ‖ 2 L $^{2}$(Ω) = ∥ P + s z ∥ 2 L 2 $_{(Ω)}$+ ∥ P − s z ∥ 2 L 2 $_{(Ω)}$+2 Re ( P + s z, P − s z ) L $^{2}$(Ω) , (3.12) 2 Re ( P + s z, P − s z ) L $^{2}$(Ω) = I$_{1}$ + I$_{2}$ + I$_{3}$ + I$_{4}$ + I$_{5}$ + I$_{6}$ + I$_{7}$ + I$_{8}$, (3.13) 7