P(ϕ)2 models with variable coefficients

We study P(φ{symbol})₂ models with variable coefficients on the nonlinear terms, for example, (□ + m²)φ{symbol} + ρ{variant}φ{symbol}³ = 0. Here p is a function on spacetime usually of the form p = p₀ + p₁ with p₀ a positive constant and p₁ small and localized. For such models we construct solutions which satisfy the CCR, have a causal dependence on p, and commute at spacelike separation. Asymptotic states and S-matrix elements are constructed when p₀ is also small. Finally, we study P(φ{symbol})₂ models on curved spacetime, for example, (g^mvτ_m{down triangle, open}_v + m²)φ{symbol} + λφ{symbol}³ = 0, and indicate a construction of local solutions.