Stationary completeness: the $N$-body short-range case

TL;DR

This paper proves that for short-range N-body Schrödinger operators, all non-threshold energies are stationary complete, ensuring strong continuity of scattering quantities and unitarity of the scattering matrix, with a new proof of asymptotic completeness.

Contribution

It establishes stationarity completeness at all non-threshold energies for short-range models, resolving a previous conjecture and improving known continuity properties.

Findings

All non-threshold energies are stationary complete.

Scattering quantities depend strongly continuously on energy.

The scattering matrix is unitary at all non-threshold energies.

Abstract

For a general class of $N$-body Schr\"odinger operators with short-range pair-potentials the wave and scattering matrices as well as the restricted wave operators are all defined at any non-threshold energy. This holds without imposing any a priori decay condition on channel eigenstates and even for models including long-range potentials of Derezi\'nski-Enss type. In this paper we improve for short-range models on the known weak continuity properties in that we show that all non-threshold energies are stationary complete, resolving in this case a conjecture from [Sk1]. A consequence is that the above scattering quantities depend strongly continuously on the energy parameter at all non-threshold energies (improving on previously almost everywhere proven properties). Another consequence is that the scattering matrix is unitary at any such energy. As a side result we obtain a new and purely stationary proof of asymptotic completeness for $N$-body short-range systems.