Green Functions and Completeness: The 3-Body Problem Revisited

Abstract 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 $$N=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . 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).