Asymptotic Behaviour Sample Clauses

Asymptotic Behaviour. ‌ We want to investigate the asymptotic behaviour when τ → ∞ of ∫ eiτψ(x;y)e—iϕ(x,ξ;y)p(x; y)σ(x, ξ)dy d¯ξ, (4.2) where ϕ is a global phase function, as in Definition 3.2, ψ ∈ C ∞(M × M) and ρ,δ α β ϕ,x x σ ∈ Sm (ϕ, T∗M). Notice that ξ y |∂ D ∂γ p(x, y)σ(x, ξ) | ≤ Cϕ,α,β,γ⟨ξ⟩m—ρ|α|+δβ. The method of stationary phase develops a technique which is useful for our purposes. In fact, the stationary phase formula provides an asymptotic expansion for integrals of the kind ∫ eiτΦ(z)a(z)dz, τ > 0, (4.3) where Φ is a real C ∞ function on Rn, a ∈ C0∞(Rn) and τ → ∞. If Φ has no critical points on the support of a, then (4.3) is rapidly decreasing as τ → ∞. Therefore, the essential contributions for the study of the asymptotic behaviour must always come from points where the phase Φ is stationary. Generally, one makes the additional hypothesis that all the critical points of Φ are non-degenerate,