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

András Vasy

Journées équations aux dérivées partielles (1999)

  • page 1-20
  • ISSN: 0752-0360

Abstract

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In these lecture notes we describe the propagation of singularities of tempered distributional solutions u ∈ 𝒮 ' of ( H - λ ) u = 0 , where H is a many-body hamiltonian H = Δ + V , Δ ≥ 0 , V = ∑ a V a , and λ 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.

How to cite

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Vasy, András. "Propagation of singularities in many-body scattering in the presence of bound states." Journées équations aux dérivées partielles (1999): 1-20. <http://eudml.org/doc/93373>.

@article{Vasy1999,
abstract = {In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u\in \mathcal \{S\}^\{\prime \}$ 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.},
author = {Vasy, András},
journal = {Journées équations aux dérivées partielles},
keywords = {wave front relation of the S-matrices},
language = {eng},
pages = {1-20},
publisher = {Université de Nantes},
title = {Propagation of singularities in many-body scattering in the presence of bound states},
url = {http://eudml.org/doc/93373},
year = {1999},
}

TY - JOUR
AU - Vasy, András
TI - Propagation of singularities in many-body scattering in the presence of bound states
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 20
AB - In these lecture notes we describe the propagation of singularities of tempered distributional solutions $u\in \mathcal {S}^{\prime }$ 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.
LA - eng
KW - wave front relation of the S-matrices
UR - http://eudml.org/doc/93373
ER -

References

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