Green functions and completeness; the $3$-body problem revisited
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929446168035328 |
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| author | Skibsted, Erik |
| author_facet | Skibsted, Erik |
| contents | Within the class of Derezi{ń}ski-Enss pair-potentials which includes Coulomb potentials a stationary scattering theory for $N$-body systems was recently developed \cite {Sk1}. In particular the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy, and this holds without imposing any a priori decay condition on channel eigenstates. In this paper we improve for the case of $3$-body systems on the known \emph{weak continuity} properties in that we show that all non-threshold energies are \emph{stationary complete} in this case, resolving a conjecture from \cite {Sk1} in the special case $N=3$. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies, hence not only almost everywhere as previously demonstrated (for an arbitrary $N$). Another consequence is that the scattering matrix is unitary at any such energy. As a side result we give an independent stationary proof of asymptotic completeness for $3$-body systems with long-range pair-potentials. This is an alternative to the known time-dependent proofs \cite{De, En}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_15028 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Green functions and completeness; the $3$-body problem revisited Skibsted, Erik Mathematical Physics Spectral Theory Within the class of Derezi{ń}ski-Enss pair-potentials which includes Coulomb potentials a stationary scattering theory for $N$-body systems was recently developed \cite {Sk1}. In particular the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy, and this holds without imposing any a priori decay condition on channel eigenstates. In this paper we improve for the case of $3$-body systems on the known \emph{weak continuity} properties in that we show that all non-threshold energies are \emph{stationary complete} in this case, resolving a conjecture from \cite {Sk1} in the special case $N=3$. A consequence is that the above scattering quantities depend \emph{strongly continuously} on the energy parameter at all non-threshold energies, hence not only almost everywhere as previously demonstrated (for an arbitrary $N$). Another consequence is that the scattering matrix is unitary at any such energy. As a side result we give an independent stationary proof of asymptotic completeness for $3$-body systems with long-range pair-potentials. This is an alternative to the known time-dependent proofs \cite{De, En}. |
| title | Green functions and completeness; the $3$-body problem revisited |
| topic | Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2205.15028 |