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We study semi-classical asymptotics for problems with localized right-hand sides by considering a Hamiltonian H (x, p) positively homogeneous of degree m1 on T^* Rⁿ0. The energy shell is H (x, p) =E, and the right-hand side fₕ is microlocalized: (1) on the vertical plane ₀=\x=x₀\; (2) on the ``cylinder'' ₀=\ (X, P) = ( (), () ) ; \ { R, () = (, ) \}. when n=2. Most precise results are obtained in the isotropic case H (x, p) =|p|ᵐ (x), with a smooth positive function. In case (2), ₀ 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.
Anikin et al. (Sun,) studied this question.