Small-data linear scattering for short-range generalized Dirac–Hartree potentials

Establish small-data linear scattering for the Dirac–Hartree equation with generalized short-range potentials |x|^{-\gamma}, where 1<\gamma<d, in both two and three spatial dimensions.

Background

The paper considers Dirac–Hartree equations with generalized potentials |x|{-\gamma} for 0<\gamma<d. A heuristic decay calculation places 1<\gamma<d in the short-range regime, because the associated nonlinear term is expected to be time-integrable, while \gamma\leq 1 is long-range and \gamma=1 is scattering-critical. The paper establishes modified scattering for the critical Coulomb potential \gamma=1 in two dimensions, but identifies small-data linear scattering for short-range potentials as unresolved in both two and three dimensions.

References

It remains an interesting open problem to establish small data linear scattering for the Dirac-Hartree equation with short range potentials, both in two and three dimensions.

— Long-range scattering for 2D Dirac-Hartree equations  (2608.12672 - Lee et al., 13 Aug 2026) in Section “Future Directions,” item (1), “Generalized potential |x|^{-\gamma}”