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 $$V_{\mu }(x)=\frac{e^{-\mu \mathrm{\mid }x\mathrm{\mid }}}{\mathrm{\mid }x\mathrm{\mid }},0\le \mu \le 1.$$ The Coulomb case corresponds to $\mu =0$, while $\mu >0$ introduces the screening length ${\mu }^{-1}$. In the limit $\mu \to 0$ and $t\to \mathrm{\infty }$, 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 $\mu t\to 0$, $\mu t\to L\in (0,\mathrm{\infty })$, or $\mu t\to \mathrm{\infty }$, with different modified scattering phases in each case.