Reducibility of Klein-Gordon equations with maximal order perturbations

We prove that all the solutions of a quasi-periodically forced linear Klein-Gordon equation $\psi_{tt}-\psi_{xx}+\mathtt{m}\psi+Q(\omega t)\psi=0$ where $Q(\omega t) := a^{(2)}(\omega t, x) \partial_{xx} + a^{(1)}(\omega t, x)\partial_x + a^{(0)}(\omega t, x)$ 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 $\omega\in {\mathbb R}^{\nu}$ 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 $Q (\omega t)$ of the second order differential operator $a^{(2)}(\omega t, x)\partial_{xx}$. 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.