Removal of the technical threshold alpha greater than one half

Prove the nonexistence of the relative wave operator $W(L_\kappa-V^L,L_\kappa)$ for smooth long-range potentials with $0<\rho_L\leq\alpha\leq1$ for all admissible $\alpha>0$, removing the current technical threshold $\alpha>1/2$ and its dimension-dependent endpoint condition.

Background

The paper establishes nonexistence of W(Lκ−VL,Lκ)W(L_\kappa-V^L,L_\kappa) for 1/2<α≤11/2<\alpha\leq1, and also for α=1/2\alpha=1/2 when d≥2d\geq2. The proof depends on propagation estimates that are not available throughout the full range 0<α≤10<\alpha\leq1.

The authors argue that the threshold is probably not a physical obstruction, especially for the massive pseudo-relativistic kinetic energy, whose group velocity is linear at small momenta.

References

We expect that the technical threshold $\alpha>1/2$ (with the endpoint $\alpha=1/2$ included when $d\geq2$) from the nonexistence Theorem~\ref{asympcomplhardyplusperturbationnonexistence}(\ref{asympcomplhardyplusperturbation:nonexistencelongrangesmallalpha}) is not necessary.

— On wave operators for Hardy operators with and without perturbations  (2610.08594 - Ishida et al., 6 Oct 2026) in Section 1, subsection “Scattering theory for more general fractional operators and Hardy potentials”