Time Global Finite-Energy Weak Solutions to the Many-Body Maxwell–Pauli Equations

Abstract

We study the quantum mechanical many-body problem of \(N \ge 1\) non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and \(K \ge 0\) 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 \(\alpha \) and the nuclear charges are not too large. The particular assumptions on the size of \(\alpha \) 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.

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Acknowledgements

This research was partially supported by US-NSF grants DMS 1600560 and DMS 1856645. Furthermore, we would like to thank Professor Michael Loss for many helpful discussions and, in particular, the proof of Lemma 1.

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Correspondence to T. F. Kieffer.

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Kieffer, T.F. Time Global Finite-Energy Weak Solutions to the Many-Body Maxwell–Pauli Equations. Commun. Math. Phys. 377, 1131–1162 (2020). https://doi.org/10.1007/s00220-020-03772-7

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