Quasiclassical asymptotics of the scattering amplitude for the scattering of a plane wave by inhomogeneities of the medium

Let $[\Delta+k^2q(x)]\psi(x,k)=0$, where $x\in\mathbf R^n$, $q(x)\in C^\infty$, $q(x)>0$, $q(x)\equiv1$ for $r=|x|>a$, and $\psi(x,k)=e^{ikx_n}+u(x,k)$, in which the function $u$ satisfies the radiation conditions
$$ u(x,k)=f(\omega,k)r^{(1-n)/2}e^{ikr}(1+O(r^{-1})),\qquad r\to\infty,\quad\omega=\frac x{r}. $$
The asymptotics of the scattering amplitude $f(\omega,k)$ for $\omega\in S^{n-1}$ is obtained as $k\to+\infty$. It can be represented in the form of a sum of two canonical operators of V. P. Maslov, constructed from the $(n-1)$-dimensional Lagrangian manifolds $L_0$, $L_+\subset T^*S^{n-1}$.
Let $\Lambda^n$ be the $n$-dimensional Lagrangian manifold comprised of the bicharacteristics corresponding to the problem under consideration, and let $s$ be a parameter along the bicharacteristics. The manifold $L_+$ can be obtained from $\Lambda^n$ by passing to spherical coordinates in $\mathbf R^{2n}_{x,p}$ projecting $\Lambda^n$ on to $T^*S^{n-1}$ and letting s go to infinity. The manifold $L_0$ coincides with $L_+$ for $q\equiv1$.
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