Semi-classical Green functions

TL;DR

This paper develops semi-classical Green functions for Hamiltonians on phase space, using Maslov's method to analyze distributions microlocalized on Lagrangian manifolds, with applications to wave beam theory.

Contribution

It introduces a semi-classical Green kernel construction for Hamiltonians with non-critical energy surfaces using Maslov's operator, advancing the understanding of wave propagation.

Findings

Construction of semi-classical Green functions satisfying the limiting absorption principle.

Application of Maslov canonical operator to microlocalized distributions.

Examples from wave beam theory illustrating the method.

Abstract

Let $H(x,p)\sim H_0(x,p)+hH_1(x,p)+\cdots$ be a semi-classical Hamiltonian on $T^*{\bf R}^n$, 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 $H^w(x,hD_x)u_h=f_h$ (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.