A Trichotomy for Modified Scattering Across the Yukawa-Coulomb Transition

We study the long-time asymptotics of the three-dimensional Hartree equation with Yukawa potential $$V_\mu(x)=\frac{e^{-\mu|x|}}{|x|}, \qquad 0\leq\mu\leq1.$$ The Coulomb case corresponds to $\mu=0$, while $\mu>0$ introduces the screening length $\mu^{-1}$. In the limit $\mu\to0$ and $t\to\infty$, the asymptotic behavior depends on the comparison between the observation scale $t$ and the screening length $\mu^{-1}$, equivalently on the parameter $\mu t$. This leads to three distinct asymptotic regimes, according as $\mu t\to0$, $\mu t\to L\in(0,\infty)$, or $\mu t\to\infty$, with different modified scattering phases in each case.