Semi-classical Green functions

Abstract : Let H(x, p) ∼ H 0 (x, p) + hH 1 (x, p) + · · · be a semi-classical Hamiltonian on T * R n , and Σ E = {H 0 (x, p) = E} a non critical energy surface. Consider f h a semi-classical distribution (the "source") microlocalized on a Lagrangian manifold Λ which intersects cleanly the flow-out Λ + of the Hamilton vector field X H 0 in Σ E. Using Maslov canonical operator, we look for a semi-classical distribution u h satisfying the limiting absorption principle and H w (x, hD x)u h = f h (semi-classical Green function). In this report, we elaborate (still at an early stage) on some results announced in [AnDoNaRo1] and provide some examples, in particular from the theory of wave beams.
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A Anikin, S. Dobrokhotov, V Nazaikinskii, Michel Rouleux. Semi-classical Green functions. International Conference on Days on Diffraction (DD), Jun 2018, Saint Petersbourg, Russia. pp.17-23, ⟨10.1109/DD.2018.8553179⟩. ⟨hal-02049796⟩



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