Asymptotic completeness for the massive pseudo-relativistic Hardy operator

Prove asymptotic completeness of the wave operator intertwining the massive pseudo-relativistic Hardy operator $(1-Delta)^{\alpha/2}-1-\kappa|x|^{-\alpha}-V^S$ and the corresponding unperturbed massive pseudo-relativistic Hardy operator $(1-\Delta)^{\alpha/2}-1-\kappa|x|^{-\alpha}$ under the short-range assumptions stated for the Fourier-multiplier class in equation (5.1), thereby covering the physically important symbol $C_0(|\xi|^2)=(|\xi|^2+1)^{\alpha/2}-1$ for $\alpha<2$.

Background

The paper extends asymptotic-completeness results from the fractional Laplacian to a class of Fourier multipliers satisfying the subordination conditions defining $K_\alpha$. It also notes that existing Kato-smoothness results apply to a broader class of symbols satisfying equation (5.1).

The massive pseudo-relativistic kinetic energy $C_0(|\xi|^2)=(|\xi|^2+1)^{\alpha/2}-1$ is physically important, but it does not satisfy the lower bound in equation (5.1) when $\alpha<2$. Consequently, the paper omits asymptotic completeness for the corresponding perturbed Hardy operator, leaving this as an explicit unresolved problem.