A Trichotomy for Modified Scattering Across the Yukawa-Coulomb Transition

Abstract

We study the long-time asymptotics of the three-dimensional Hartree equation with Yukawa potential \[ V_μ(x)=\frac{e^{-μ|x|}}{|x|}, \qquad 0\leqμ\leq1. \] The Coulomb case corresponds to $\mu =0$, while $\mu >0$ introduces the screening length ${\mu }^{-1}$. In the limit $\mu \to 0$ and $t\to \infty$, 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 $\mu t\to 0$, $\mu t\to L\in (0,\infty )$, or $\mu t\to \infty$, with different modified scattering phases in each case.