Stationary completeness: The N-body short-range case

TY - JOUR

T1 - Stationary completeness

T2 - The N-body short-range case

AU - Skibsted, E.

PY - 2025/12

Y1 - 2025/12

N2 - For a general class of N-body Schrödinger 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 Dereziński-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 [21]. 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.

AB - For a general class of N-body Schrödinger 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 Dereziński-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 [21]. 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.

KW - Minimum generalized eigenfunctions

KW - N-body Schrödinger operators

KW - Scattering and wave matrices

KW - Stationary short-range scattering theory

UR - https://www.scopus.com/pages/publications/105016389384

U2 - 10.1016/j.aim.2025.110544

DO - 10.1016/j.aim.2025.110544

M3 - Journal article

AN - SCOPUS:105016389384

SN - 0001-8708

VL - 481

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 110544

ER -