In Beig and Simon (Commun Math Phys 144:373–390, 1992) the authors prove a uniqueness theorem for static solutions of the Einstein–Euler system which applies to fluid models whose equation of state fulfills certain conditions. In this article it is shown that the result of Beig and Simon (1992) can be applied to isotropic Vlasov matter if the gravitational potential well is shallow. To this end we first show how isotropic Vlasov matter can be described as a perfect fluid giving rise to a barotropic equation of state. This Vlasov equation of state is investigated, and it is shown analytically that the requirements of the uniqueness theorem are met for shallow potential wells. Finally the regime of shallow gravitational potential is investigated by numerical means. An example for a unique static solution is constructed, and it is compared to astrophysical objects like globular clusters. Finally we find numerical indications that solutions with deep potential wells are not unique.

中文翻译：

---

爱因斯坦-弗拉索夫系统静态各向同性低压解的唯一性

在 Beig 和 Simon (Commun Math Phys 144:373–390, 1992) 中，作者证明了爱因斯坦-欧拉系统静态解的唯一性定理，该定理适用于状态方程满足特定条件的流体模型。本文表明，如果引力势阱较浅，Beig 和 Simon (1992) 的结果可以应用于各向同性的 Vlasov 物质。为此，我们首先展示了如何将各向同性的弗拉索夫物质描述为产生正压状态方程的完美流体。研究了这个 Vlasov 状态方程，分析表明对于浅势阱满足唯一性定理的要求。最后通过数值方法研究了浅层引力势的规律。构建了一个唯一静态解决方案的示例，并将其与球状星团等天体物理物体进行比较。最后，我们发现数值指示表明深位势井的解不是唯一的。