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 $μ=0$, while $μ&gt;0$ introduces the screening length $μ<sup>{-1}$. In the limit $μ\to0$ and $t\to\infty$, the asymptotic behavior depends on the comparison between the observation scale $t$ and the screening length $μ<sup>{-1}$, equivalently on the parameter $μt$. This leads to three distinct asymptotic regimes, according as $μt\to0$, $μt\to L\in(0,\infty)$, or $μt\to\infty$, with different modified scattering phases in each case.

• The paper establishes a trichotomy in asymptotic behavior, rigorously differentiating Coulomb, intermediate, and Yukawa scattering regimes.
• It employs uniform dispersive decay and weighted Sobolev techniques to derive explicit asymptotic profiles and phase corrections.
• The study reveals non-commutativity of the limits μ → 0 and t → ∞, highlighting the role of screening in long-range interactions.

Trichotomy for Modified Scattering in the Hartree Equation: The Yukawa–Coulomb Transition

Problem Formulation and Historical Context

This work investigates the long-time asymptotic dynamics of the three-dimensional Hartree equation with Yukawa or Coulomb-type nonlocal nonlinearities. The model is given by

$$\partial_t u_\mu = -\frac12\Delta u_\mu + \kappa\,(V_\mu*u_\mu^2)u_\mu,\quad u_\mu(0)=u_\mathrm{in},$$

where $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$, with $0\leq\mu\leq1$. The parameter $\mu$ tunes the interaction from long-range Coulomb ($\mu=0$) to exponentially decaying Yukawa potentials ($\mu>0$). This equation arises, for example, as a mean-field model with screened Coulomb (Yukawa) interactions, e.g., in the Thomas–Fermi approximation in electronic structure theory.

Historically, the scattering theory for such equations splits sharply between short-range perturbations (Yukawa) and the truly long-range Coulomb case. In the latter, classical scattering fails, and one must employ modified scattering with a nontrivial, typically logarithmic, phase correction. The work of Ginibre–Ozawa, Hayashi–Naumkin, and others clarified that the Coulomb potential generates a phase shift growing like $\log t$ at long times. In contrast, Yukawa potentials (for fixed $\mu>0$) are genuinely short-range and admit standard scattering asymptotics.

Main Results: A Dynamical Trichotomy

The principal contribution is the precise mathematical description of how asymptotic states transition from the Coulomb to the Yukawa regime as $\mu\to0$, especially when the observation time $t$ and screening length $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$0 simultaneously diverge. Crucially, the order in which these limits are taken matters, leading to non-commutativity of the Coulomb and large-time limits.

The key parameter is $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$1, the ratio of the observation time to the screening length. The asymptotic behavior of small-data solutions (in a weighted Sobolev space $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$2) then exhibits a trichotomy, depending on the limiting behavior of $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$3 along sequences $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$4, $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$5:

1. Coulomb-like ($V_\mu(x) = |x|^{-1}e^{-\mu|x|}$6):
   • The solution displays the standard Coulomb modified scattering with a logarithmic phase correction $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$7.
2. Transition ($V_\mu(x) = |x|^{-1}e^{-\mu|x|}$8):
   • The asymptotics have a finite, explicit phase correction depending on $V_\mu(x) = |x|^{-1}e^{-\mu|x|}$9 (the product of the small Yukawa parameter and large time), interpolating between the Coulomb and Yukawa cases.
3. Yukawa-like ($0\leq\mu\leq1$0):
   • The logarithmic phase shift saturates at $0\leq\mu\leq1$1, and an additional limiting Yukawa correction emerges, reflecting the dominance of short-range screening.

These regimes are captured by an explicit, unitarily equivalent asymptotic formula for $0\leq\mu\leq1$2 as $0\leq\mu\leq1$3: $0\leq\mu\leq1$4 where the structure and scaling of the phase $0\leq\mu\leq1$5 differ in each regime (Coulomb, transition, Yukawa). The precise forms of the phase functionals—the Coulomb component, transition kernel, and Yukawa limit (involving the exponential integral)—are explicitly computed.

Technical Analysis

The methodology couples a rigorous phase-removal analysis in self-similar variables with uniform-in-$0\leq\mu\leq1$6 dispersive decay and weighted Sobolev control, extending the Hayashi–Naumkin weighted-profile argument to hold uniformly across $0\leq\mu\leq1$7. By rescaling to an amplitude $0\leq\mu\leq1$8 in velocity variables, the nonlinear evolution reduces (after removing the long-range phase) to an integrable perturbation, producing strong convergence in $0\leq\mu\leq1$9 to a profile $\mu$0.

A critical aspect is tracking the dependence of the effective scattering phase on both $\mu$1 and $\mu$2. The analysis highlights the non-commutation of limits $\mu$3 and $\mu$4: taking $\mu$5 before $\mu$6 recovers the Coulomb logarithmic phase, while the reversed order yields a time-saturation effect. The switching of regimes is identified by precise norm estimates and continuity properties of the limiting profiles and phase operators as $\mu$7, including detailed quantitative bounds showing that the difference in asymptotic profiles multiplied by the logarithmic phase vanishes: $\mu$8

Strong numerical results include:

• Uniform global-in-time decay $\mu$9 for all $\mu=0$0.
• Explicit rates of convergence for the difference between Yukawa and Coulomb asymptotics in the critical window $\mu=0$1.

Consequences and Implications

This work rigorously demonstrates that there is no continuous Coulomb → Yukawa transition in the long-range scattering phase: the asymptotic behavior bifurcates into three sharply differentiated regimes depending on the interaction between time and the screening scale. This answers a subtle question about the physical/statistical mechanical interpretation of screening in mean-field equations: at sufficiently large times, the effective phase memory of the system can transition from long-range Coulomb to short-range Yukawa, but only after a finite screening-length-dependent saturation.

In addition to the explicit trichotomy theorem, the paper provides a technical template for analyzing two-parameter limits in dispersive PDEs with slowly-decaying nonlocal nonlinearities. The uniformity of the analytic estimates with respect to the screening parameter is essential both for the clarity of the limiting process and for applications in many-body physics where screening is weak but nonzero.

Another noteworthy implication is the possibility (suggested for future work) to relax the weighted Sobolev small-data assumption using wave-packet or physical-space energy dispersion techniques, potentially bringing the mathematical theory closer to physically relevant settings.

Future Directions

Possible extensions include:

• Generalization to other models (non-Hartree nonlocalities, higher-order nonlinearities).
• Development of wave-packet-based approaches to further weaken regularity or smallness conditions.
• Application of the trichotomy framework to more complex mean-field or kinetic equations with competing interactions and multiple scales.

The non-commutativity of the large-time/$\mu=0$2 limits observed here may also have analogues in other singular perturbation problems for long-range, weakly screened dispersive systems.

Conclusion

The paper establishes a mathematically sharp description of the transition between Coulomb and Yukawa modified scattering in the 3D Hartree equation, demonstrating a structurally stable trichotomy for the limiting behavior as both the screening parameter and observation time diverge. This clarifies and quantifies the nontrivial interplay between long-range dispersion and screening, identifying precise asymptotic phases and strong uniform decay properties across the transition. The techniques and results are of substantial value for the rigorous analysis of asymptotic dynamics in nonlocal nonlinear dispersive PDEs.