2025/01/03 by Rouleux, Michel
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2501.02079
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^*\bf Rn∖0. The energy shell is H(x,p)=E, and the right-hand side fh is microlocalized: (1) on the vertical plane Λ0=\x=x0\; (2) on the ``cylinder'' Λ0=\(X,P)=(φω(ψ),ω(ψ)); φ∈ \bf R, ω(ψ)=(cosψ,sinψ)\. when n=2. Most precise results are obtained in the isotropic case H(x,p)=|p|m\overρ(x), with ρ a smooth positive function. In case (2), Λ0 is the frequency set of Bessel function J0(|x|\over 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 \it bi-canonical operator.