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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.02079 |
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| _version_ | 1866909447945715712 |
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| author | Rouleux, Michel |
| author_facet | Rouleux, Michel |
| contents | We study semi-classical asymptotics for problems with localized right-hand sides by considering a Hamiltonian $H(x,p)$ positively homogeneous of degree $m\geq1$ on $T^*{\bf R}^n\setminus0$. The energy shell is $H(x,p)=E$, and the right-hand side $f_h$ is microlocalized: (1) on the vertical plane $Λ_0=\{x=x_0\}$; (2) on the ``cylinder'' $Λ_0=\{(X,P)=\bigl(φω(ψ),ω(ψ)\bigr); \ φ\in {\bf R}, ω(ψ)=(\cosψ,\sinψ)\}$. 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), $Λ_0$ is the frequency set of Bessel function $J_0({|x|\over h})$, and the solution $u_h$ of $(H(x,hD_x)-E)u_h=f_h$ when $m=1$, already provides an insight in the structure of ``Bessel beams'', which arise in the theory of optical fibers. We present 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}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_02079 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semiclassical Green functions and Lagrangian intersection. Applications to the propagation of Bessel beams in non-homogeneous media Rouleux, Michel Analysis of PDEs Mathematical Physics We study semi-classical asymptotics for problems with localized right-hand sides by considering a Hamiltonian $H(x,p)$ positively homogeneous of degree $m\geq1$ on $T^*{\bf R}^n\setminus0$. The energy shell is $H(x,p)=E$, and the right-hand side $f_h$ is microlocalized: (1) on the vertical plane $Λ_0=\{x=x_0\}$; (2) on the ``cylinder'' $Λ_0=\{(X,P)=\bigl(φω(ψ),ω(ψ)\bigr); \ φ\in {\bf R}, ω(ψ)=(\cosψ,\sinψ)\}$. 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), $Λ_0$ is the frequency set of Bessel function $J_0({|x|\over h})$, and the solution $u_h$ of $(H(x,hD_x)-E)u_h=f_h$ when $m=1$, already provides an insight in the structure of ``Bessel beams'', which arise in the theory of optical fibers. We present 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}. |
| title | Semiclassical Green functions and Lagrangian intersection. Applications to the propagation of Bessel beams in non-homogeneous media |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2501.02079 |