Semi-Classical Green Functions and Lagrangian Intersection. Applications to the Propagation of Bessel Beams in Non-Homogeneous Media

Abstract

We study semi-classical asymptotics for problems with localized right-hand sides by considering a Hamiltonian $H(x,p)$ positively homogeneous of degree $m\ge 1$ on $T^{*}{\mathbf{R}}^n\setminus 0$ . The energy shell is $H(x,p)=E$ , and the right-hand side $f_h$ is microlocalized: (1) on the vertical plane ${\mathrm{\Lambda }}_0=\{x=x_0\}$ ; (2) on the “cylinder” ${\mathrm{\Lambda }}_0=\{(X,P)=(\phi \omega (\psi ),\omega (\psi ));\phi \in \mathbf{R},\omega (\psi )=(\cos \psi ,\sin \psi )\}$ , when $n=2$ . Most precise results are obtained in the isotropic case $H(x,p)=|p{|}^m/\rho (x)$ , with $\rho$ a smooth positive function. In case (2), ${\mathrm{\Lambda }}_0$ is the frequency set of Bessel function $J_0(|x|/h)$ , and the solution $u_h$ of $(H(x,h{D}_x)-E)u_h=f_h$ when $m=1$ , already provides an insight in the structure of “Bessel beams,” which arise in the theory of optical fibers. We present some extensions of our previous works with A. Anikin, S. Dobrokhotov and V. Nazaikinskii. In Section 3, we sketch the semi-classical counterpart of the construction of parametrices for the Cauchy problem with Lagrangian intersections, as is set up by Melrose and Uhlmann. This involves Maslov bi-canonical operator.