Semiclassical Green functions and Lagrangian intersection. Applications to the propagation of Bessel beams in non-homogeneous media

A. Yu. Anikin , S Dobrokhotov , Vladimir Nazaikinskii , Michel L. Rouleux (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E8 Dynamique quantique et analyse spectrale

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 * R^2 \ 0. The energy shell is E = 1, and the right-hand side f_h is microlocalized: (1) on the vertical plane Λ = {x = x_0 }; (2) on the "cylinder" Λ = {(X, P) = ϕω(ψ), ω(ψ) ; ϕ ∈ R, ω(ψ) = (cos ψ, sin ψ)}. We restrict essentially to the isotropic case H(x, p) = |p|^m/ ρ(x) , with ρ a smooth positive function. In case (2), Λ is the frequency set of Bessel function J_0 (|x|/h), and the solution u_h of (H(x, hD_x) − 1)u_h = f _h , which is called a "Bessel beam", arises in the theory of optical fibers.

Mots clés: Semi-classical analysis; Maslov theory; eikonal coordinates

Domaines: Mathématiques [math]

A. Yu. Anikin, S Dobrokhotov, Vladimir Nazaikinskii, Michel L. Rouleux. Semiclassical Green functions and Lagrangian intersection. Applications to the propagation of Bessel beams in non-homogeneous media. 2021. ⟨hal-02973891v3⟩