$ close to $\Phi_0 $ in $C^\infty $ and we make the corresponding replacement of $\Psi _0$. Also recall that because of the strict pluri-subharmonicity of $\Phi $, we have [$$\label{ev.0.4}
2\Re \Psi (x,\overline{y})-\Phi (x)-\Phi (y)\asymp -|x-y|^2,$$]{} so the uniform boundedness $H_\Phi \to H_\Phi $ follows from the domination of the modulus of the effective kernel by a Gaussian convolution kernel.

Next, consider the evolution problem [$$\label{ev.0.5}
(h\partial _t+\widetilde{P})\widetilde{U}(t)=0,\ \widetilde{U}(0)=1,$$]{} where $t$ is restricted to the interval $[0,h^\delta ]$ for some arbitrarily small but fixed $\delta >0$. We review how to solve this problem approximately by a geometrical optics construction: Look for $\widetilde{U}(t)$ of the form [$$\label{ev.0.6}
\widetilde{U}(t)u(x)=h^{-n}{\int\hskip -2mm\int}e^{\frac{2}{h}\Psi
  _t(x,\overline{y})}
a_t(x,\overline{y};h) u(y) e^{-2\Phi _0(y)/h}dyd\overline{y},$$]{} where $\Psi _t$, $a_t$ depend smoothly on all the variables and $\Psi _{t=0}=\Psi _0$, $a _{t=0}=a _0$ in (\[ev.0.3\]), so that $\widetilde{U}(0)=1$ up to a negligible operator.

Notice that formally $\widetilde{U}(t)$ is the Fourier integral operator [$$\label{ev.0.7}
\widetilde{U}(t)u(x)=h^{-n}{\int\hskip -2mm\int}e^{\frac{2}{h}(\Psi _t(x,\theta
  )-\Psi _0(y,\theta ))}a_t(x,\theta ;h)u(y) dyd\theta ,$$]{} where we choose the integration contour $\theta =\overline{y}$. Writing $2\Psi _t(x,\theta )=i\phi _t(x,\theta )$ leads to more customary notation and we impose the eiconal equation [$$\label{ev.0.8}
i\partial _t\phi +\widetilde{p}(x,\phi '_x(x,\theta ))=0.$$]{} Of course, we are manipulating $C^\infty $ functions in 