András Vasy
February 17, 1999
In this talk I will extend the familiar example that singularities of solutions of the wave equation propagate along (generalized) broken bicharcteristics, to many-body scattering. Such a description connects geometric optics, where light is modelled by little billiard balls, moving in straight lines and reflecting from surfaces according to Snell's law, and wave phenomena. In this talk I will discuss an analogous connection between quantum and classical many-body systems.
Let H be a many-body Hamiltonian, so 
, 
,
where the two-body interactions 
are real-valued
polyhomogeneous symbols of order -1 (e.g. Coulomb-type with the singularity
at the origin removed).
I will discuss the propagation of singularities of generalized
eigenfunctions of H, where `singularities' are understood as
a description of the lack of
decay at infinity.
In particular, I will explain the
propagation results in three-body scattering, and also in many-body
scattering under the additional assumption
that no subsystem has a bound state.
These results provide a connection between
quantum mechanics and classical mechanics which is very similar to the
one between wave propagation and geometric optics discussed above.
They also prove that
the wave front relation of the free-to-free S-matrix (which
is all of the S-matrix if no subsystem has a bound state)
is given by the broken geodesic
flow, broken at the `singular directions' corresponding to
the collision planes, on 
at time 
.
In many-body scattering with at least four particles, if some subsystems
have bound states,
one would have to combine the behavior of bound states with the
classical dynamics to understand the propagation of singularities.
These results have natural geometric generalizations on asymptotically Euclidean spaces X with Hamiltonians that are singular at certain submanifolds of the geodesic compactification of X.