Propagation of singularities in many-body scattering in the presence of bound states

Vasy, András

Journées équations aux dérivées partielles (1999), article no. 16, 20 p.

In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u\in {𝒮}^{\text{'}}$ of $(H-\lambda )u=0$, where $H$ is a many-body hamiltonian $H=\Delta +V$, $\Delta \ge 0$, $V={\sum }_a{V}_a$, and $\lambda$ is not a threshold of $H$, under the assumption that the inter-particle (e.g. two-body) interactions $V_a$ are real-valued polyhomogeneous symbols of order $-1$ (e.g. Coulomb-type with the singularity at the origin removed). Here the term “singularity” provides a microlocal description of the lack of decay at infinity. Our result is then that the set of singularities of $u$ is a union of maximally extended broken bicharacteristics of $H$. These are curves in the characteristic variety of $H$, which can be quite complicated due to the existence of bound states. We use this result to describe the wave front relation of the S-matrices. Here we only present the statement of the results and sketch some of the ideas in proving them, the complete details will appear elsewhere.

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