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

We study semi-classical asymptotics for problems with localized right-hand sides by considering a Hamiltonian $H(x,p)$ positively homogeneous of degree m≥1 on T^* Rⁿ0. The energy shell is $H(x,p)=E$, and the right-hand side fₕ is microlocalized: (1) on the vertical plane Λ₀==x₀\; (2) on the ``cylinder'' Λ₀=\(X,P)=(φω(ψ),ω(ψ)); \ φ∈ R, ω(ψ)=(cosψ,sinψ)\. when $n=2$. Most precise results are obtained in the isotropic case H(x,p)=|p|ᵐρ(x), with ρ a smooth positive function. In case (2), Λ₀ is the frequency set of Bessel function J₀(|x| h), and the solution uₕ of (H(x,hDₓ)-E)uₕ=fₕ when $m=1$, already provides an insight in the structure of ``Bessel beams'', which arise in the theory of optical fibers. We present in this work some extensions of A.Anikin, S.Dobrokhotov, V.Nazaikinskii, M.Rouleux, Theor. Math. Phys. 214(1): p.1-23, 2023. In Sect.3 we sketch the semi-classical counterpart of the construction of parametrices for the Cauchy problem with Lagrangian intersections, as is set up by R.Melrose and G.Uhlmann. This involves Maslov { bi-canonical operator}.