Reducibility of Klein-Gordon equations with maximal order perturbations

We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation ψₜₜ-ψₓₓ+mψ+Q(ω t)ψ=0 where Q(ω t) := a⁽²⁾(ω t, x) ∂ₓₓ + a⁽¹⁾(ω t, x)∂ₓ + a⁽⁰⁾(ω t, x) is a differential operator of order $ 2 $, parity preserving and reversible, are almost periodic in time and uniformly bounded for all times, provided that the coefficients a⁽²⁾, a⁽¹⁾, a⁽⁰⁾ are small enough and the forcing frequency ω∈ Rν belongs to a Borel set of asymptotically full measure. This result is obtained by reducing the Klein-Gordon equation to a diagonal constant coefficient system with purely imaginary eigenvalues. The main difficulty is the presence in the perturbation Q (ω t) of the second order differential operator a⁽²⁾(ω t, x)∂ₓₓ. In suitable coordinates the Klein-Gordon equation is the composition of two backward/forward quasi-periodic in time perturbed transport equations with non-constant coefficients, up to lower order pseudo-differential remainders. A key idea is to straighten this first order pseudo-differential operator with bi-characteristics through a novel quantitative Egorov analysis.