Analyticity of scattering for the ϕ⁴ theory

Abstract

We consider scattering for the equation (□+m²)ϕ+λϕ³=0 on four-dimensional Minkowski space. Form>0, one-to-one and onto wave operatorsW _± ^λ :H→H are known to exist for all λ≧0, whereH denotes the Hilbert space of finite-energy Cauchy data. We prove that the maps (λ,u)↦W _± ^λ (u) and (λ,u)↦(W _± ^λ )⁻¹ (u) are continuous from [0, ∞)×H toH, and extend to real-analytic functions from an open neighborhood of {0}×H∪ℝ×{0}⊂ℝ×H to the Hilbert spaceH₋₁ of Cauchy data with Poincaré-invariant norm. Form=0, wave operatorsW _± ^λ are known to exist as diffeomorphisms ofH for all λ≧0, where hereH denotes the Hilbert space of finite Einstein energy Cauchy data. In this case we prove that the maps (λ,u)↦(W _± ^λ ) (u) and (λ,u)↦(W _± ^λ )⁻¹ (u) extend to real-analytic functions from a neighborhood of [0, ∞)×H⊂ℝ×H toH.