Asymptotic properties of non-relativistic limit for pseudo-relativistic Hartree equations
Pan Chen, Vittorio Coti Zelati, Yuanhong Wei
TL;DR
This work analyzes the non-relativistic limit of the pseudo-relativistic Hartree equation on $\mathbb{R}^3$ as $c \to \infty$, focusing on both energy and action ground states and their variational characterizations. It develops a rigorous framework to prove the convergence $u_c \to u_\infty$ in $H^s$ for all $s$, and constructs higher-order corrections $f_j$ by solving linearized problems $\mathcal{L} f_j = -\sum_{\cdots} + \mathcal{T}_n$, yielding precise asymptotic expansions $c^{2n} u_c \approx \sum_{j=0}^{n} c^{2(n-j)} f_j$. For the action ground state, the paper proves a full multi-term expansion with explicit recursion and solvability conditions, governed by the invertibility of the linearized operator $\mathcal{L}$. For the energy ground state, it establishes analogous expansions, derives the asymptotics of the Lagrange multipliers and energy, and identifies the leading correction through $\mathcal{L} f_1 = -\frac{1}{8 m^3} (-\Delta)^2 u_\infty$. Together, these results provide a detailed, higher-order understanding of the pseudo-relativistic to non-relativistic transition in mean-field Hartree dynamics, extending prior action-state results to energy-state asymptotics and clarifying the role of linearized-inhomogeneous corrections.
Abstract
In this paper, we study the asymptotic behavior of energy and action ground states to the following pseudo-relativistic Hartree equation \[ \left(\sqrt{-c^2Δ+m^2c^4}-mc^2\right)u + λu = \left(|x|^{-1}*|u|^2\right)u \] as the speed of light $c\to\infty$. We obtain an asymptotic expansion of the ground state as $c \to \infty,$ which is new in the case of the energy ground state and generalizes the results of Choi, Hong, and Seok (2018) for the action ground state.