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<p>Within the class of Dereziński–Enss pair-potentials which includes Coulomb potentials, a stationary scattering theory for <i>N</i>-body systems was recently developed (Skibsted in Commun Math Phys 401:2193–2267, 2023). 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 <i>weak continuity</i> properties in that we show that all non-threshold energies are <i>stationary complete</i> in this case, resolving a conjecture from Skibsted (Commun Math Phys 401:2193–2267, 2023) in the special case <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="23_2025_1629_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(N=3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. A consequence is that the above scattering quantities depend <i>strongly continuously</i> on the energy parameter at all non-threshold energies, hence not only almost everywhere as previously demonstrated (for an arbitrary <i>N</i>). Another consequence is the unitary of the scattering matrix 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 (Dereziński in Ann Math 38:427–476, 1993; Enss in Long-range scattering of two- and three-body quantum systems, Publications ecole polytechnique, Palaiseau, 1989).</p>

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Green Functions and Completeness: The 3-Body Problem Revisited

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  • 发表日期：

    2025年10月7日

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  • E. Skibsted

摘要

Within the class of Dereziński–Enss pair-potentials which includes Coulomb potentials, a stationary scattering theory for N-body systems was recently developed (Skibsted in Commun Math Phys 401:2193–2267, 2023). 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 weak continuity properties in that we show that all non-threshold energies are stationary complete in this case, resolving a conjecture from Skibsted (Commun Math Phys 401:2193–2267, 2023) in the special case \(N=3\) N = 3 . A consequence is that the above scattering quantities depend 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 the unitary of the scattering matrix 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 (Dereziński in Ann Math 38:427–476, 1993; Enss in Long-range scattering of two- and three-body quantum systems, Publications ecole polytechnique, Palaiseau, 1989).