We study the quantum mechanical many-body problem of $$N \ge 1$$ N ≥ 1 non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $$K \ge 0$$ K ≥ 0 static nuclei. We model the dynamics of the electrons and their self-generated electromagnetic field with the so-called many-body Maxwell–Pauli equations. Here we construct time global, finite-energy, weak solutions to the many-body Maxwell–Pauli equations under the assumption that the fine structure constant $$\alpha $$ α and the nuclear charges are not too large. The particular assumptions on the size of $$\alpha $$ α and the nuclear charges ensure that we have energetic stability of the many-body Pauli Hamiltonian, i.e., the ground state energy is finite and uniformly bounded below with lower bound independent of the magnetic field and the positions of the nuclei. This work serves as an initial step towards understanding the connection between the energetic stability of matter and the well-posedness of the corresponding dynamical equations.

中文翻译：

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多体麦克斯韦-泡利方程的时间全局有限能量弱解

我们研究了 $$N \ge 1$$ N ≥ 1 非相对论电子的量子力学多体问题，其自旋与其自产生的经典电磁场相互作用，以及 $$K \ge 0$$ K ≥ 0 静态核. 我们使用所谓的多体麦克斯韦-泡利方程对电子的动力学及其自生电磁场进行建模。在这里，我们在精细结构常数 $$\alpha $$ α 和核电荷不太大的假设下，构造了多体麦克斯韦-泡利方程的时间全局、有限能量、弱解。$$\alpha $$ α 的大小和核电荷的特定假设确保我们具有多体泡利哈密顿量的能量稳定性，即，基态能量是有限的，并且下界与磁场和原子核的位置无关。这项工作是了解物质能量稳定性与相应动力学方程适定性之间联系的第一步。