Propagation of singularities in three-body scattering. (English) Zbl 0941.35001

Astérisque. 262. Paris: Société Mathématique de France. iv, 151 p. (2000).
In this memoire the author considers a compact manifold with boundary \(X\) equipped with a scattering metric \(g\) and with a collection \(c_i\) of disjoint closed embedded submanifolds of \(\partial X\). Let \(\Delta\) be the Laplacian of \(g\), suppose that \(V\in C^\infty ([X_i\cup_iC_i])\) where \([X_i \cup_iC_i]\) is \(X\) blown up along \(C_i\), assume that \(V\) vanishes at the lift of \(\partial X\), and consider the operator \(H=\Delta+V\). The author analyzes the propagation of singularities of generalized eigenfunctions of \(H\), showing that this is essentially a hyperbolic problem which has much in common with the Dirichlet and transmission problems for the wave operator. The author shows also that the wave front relation of the free-to-free part of the scattering matrix is given by the broken geodesic flow at distance \(\pi\).
Reviewer: N.Jacob (Erlangen)


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35P25 Scattering theory for PDEs
81U10 \(n\)-body potential quantum scattering theory