Semi-classical Green functions

Abstract

Let $H(x,p)\sim H_0(x,p)+h{H}_1(x,p)+\cdots$ be a semi-classical Hamiltonian on $T^{\ast }{\mathbf{R}}^n$, and ${\Sigma }_E=\{H_0(x,p)=E\}$ a non critical energy surface. Consider $f_h$ a semi-classical distribution (the "source") microlocalized on a Lagrangian manifold $\Lambda$ which intersects cleanly the flow-out ${\Lambda }_{+}$ of the Hamilton vector field $X_{H_0}$ in ${\Sigma }_E$. Using Maslov canonical operator, we look for a semi-classical distribution $u_h$ satisfying the limiting absorption principle and $H^w(x,h{D}_x)u_h=f_h$ (semi-classical Green kernel). In this report, we elaborate (still at an early stage) on some results announced in [Doklady Akad. Nauk, Vol. 76, No1, p.1-5, 2017] and provide some examples, in particular from the theory of wave beams.

Conference Days of Diffraction 2018, St Petersburg