Propagation Estimates for the Boson Star Equation

TL;DR

This work studies propagation for the boson-star equation $i\partial_t\psi=(\langle\nabla\rangle+w*|\psi|^2)\psi$ in $\mathbb{R}^d$, addressing how fast the quantum state can disperse for general two-body potentials $w$ decomposed as a finite signed measure plus a bounded function. It develops a comprehensive well-posedness theory (local and global) in varied function spaces (including Lorentz and weighted Sobolev spaces) and proves sharp maximal-velocity bounds that match the light-speed limit with exponentially small remainders. In the short-range setting, the paper establishes scattering to free dynamics, wave-operator invertibility in weighted spaces, and asymptotic phase-space propagation and minimal-velocity estimates, linking long-time behavior to the momentum of scattering states. The results provide a rigorous, quantitative picture of propagation and dispersive decay for pseudo-relativistic boson stars, with implications for mean-field limits and gravitational models of boson stars. The methods combine fixed-point, dispersive, Lorentz-space inequalities, and modern ASTLO/phase-space techniques to handle nonlinear, nonlocal interactions and time-dependent effective Hamiltonians. The work thus advances understanding of causality-like propagation and long-time scattering in nonlinear, nonlocal quantum dynamics.

Abstract

We consider the boson star equation with a general two-body interaction potential $w$ and initial data $ψ_0$ in a Sobolev space. Under general assumptions on $w$, namely that $w$ decomposes as a sum of a finite, signed measure and an essentially bounded function, we prove that the (local in time) solution cannot propagate faster than the speed of light, up to a sharp exponentially small remainder term. If $w$ is short-range and $ψ_0$ is regular and small enough, we prove in addition asymptotic phase-space propagation estimates and minimal velocity estimates for the (global in time) solution, depending on the momentum of the scattering state associated to $ψ_0$.

Figures (3)

• Figure 1: Dimension $d\geq 4$. Well-Posedness of \ref{['eq:hartree_semi_intro']}: Admissible class of $w$ depending on the regularity $s$ of $\psi_0$. A thick vertical line with extremities at $1/q_1$ and $1/q_2$ represents the space $L^{q_1,\tilde{q}_1}+L^{q_2,\tilde{q}_2}$, with $\tilde{q}_j=q_j$ if the extremity is a disk ($\bullet$), $\tilde{q}_j=\infty$ if the extremity is diamond (origin=c]45$\blacksquare$). There are two special cases: the $L^{q,\tilde{q}}$ space is replaced by $\mathcal{M}$ if the extremity is a square ($\blacksquare$), and by $\bigcup_{q>q_j}L^q$ if it is a circle ($\boldsymbol{\circ}$). Solid lines correspond to global well-posedness while dotted lines correspond to local well-posedness.
• Figure 2: Dimension $d=3$. Well-Posedness of \ref{['eq:hartree_semi_intro']}: Admissible class of $w$ depending on the regularity $s$ of $\psi_0$. A thick vertical line with extremities at $1/q_1$ and $1/q_2$ represents the space $L^{q_1,\tilde{q}_1}+L^{q_2,\tilde{q}_2}$, with $\tilde{q}_j=q_j$ if the extremity is a disk ($\bullet$), $\tilde{q}_j=\infty$ if the extremity is diamond (origin=c]45$\blacksquare$). There are two special cases: the $L^{q,\tilde{q}}$ space is replaced by $\mathcal{M}$ if the extremity is a square ($\blacksquare$), and by $\bigcup_{q>q_j}L^q$ if it is a circle ($\boldsymbol{\circ}$). Solid lines correspond to global well-posedness while dotted lines correspond to local well-posedness.
• Figure 3: Global Well-Posedness in the Short range case: The green region represents the domain of admissible pairs $(\frac{1}{p'},\frac{1}{q})$ such that, if $\psi_0$ is in $H^s\cap H^{s,p'}$ and $w$ in $L^q+\mathcal{M}$ and their norms are sufficiently small, then \ref{['eq:hartree_semi_intro']} admits a global solution, provided by Theorem \ref{['th:global-short-range-intro']}.

Theorems & Definitions (97)

• Remark 2.1
• Theorem 2.2: Local existence I
• Theorem 2.3: Local existence II
• Theorem 2.4: Sharp maximal velocity estimate for convex subsets
• Remark 2.5
• Proposition 2.6: Maximal velocity estimate for general subsets
• Theorem 2.7: Global existence for long-range interaction potentials I
• Remark 2.8
• Theorem 2.9: Global existence for long-range interaction potentials II
• Theorem 2.10: Global existence for short-range interaction potentials