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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Fröhlich, J., Lieb, E.H., Loss, M.: Stability of Coulomb systems with magnetic fields. I. The one-electron atom. Commun. Math. Phys. 104(2), 251–270 (1986)
Lieb, E.H., Loss, M.: Stability of Coulomb systems with magnetic fields. II. The many-electron atom and the one-electron molecule. Commun. Math. Phys. 104(2), 271–282 (1986)
Loss, M., Yau, H.-T.: Stabilty of Coulomb systems with magnetic fields. III. Zero energy bound states of the Pauli operator. Commun. Math. Phys. 104(2), 283–290 (1986)
Fefferman, C.: Stability of coulomb systems in a magnetic field. Proc. Natl. Acad. Sci. 92(11), 5006–5007 (1995)
Lieb, E.H., Loss, M., Solovej, J.P.: Stability of matter in magnetic fields. Phys. Rev. Lett. 75, 985–989 (1995)
Nakamitsu, K., Tsutsumi, M.: The Cauchy problem for the coupled Maxwell–Schrödinger equations. J. Math. Phys. 27(1), 211–216 (1986)
Tsutsumi, Y.: Global existence and asymptotic behavior of solutions for the Maxwell–Schrödinger equations in three space dimensions. Commun. Math. Phys. 151(3), 543–576 (1993)
Guo, Y., Nakamitsu, K., Strauss, W.: Global finite-energy solutions of the Maxwell–Schrödinger system. Commun. Math. Phys. 170(1), 181–196 (1995)
Ginibre, J., Velo, G.: Long range scattering and modified wave operators for the Maxwell–Schrödinger system. I. The case of vanishing asymptotic magnetic field. Commun. Math. Phys. 236(3), 395–448 (2003)
Shimomura, A.: Modified wave operators for Maxwell–Schrödinger equations in three space dimensions. Ann. Henri Poincaré 4(4), 661–683 (2003)
Ginibre, J., Velo, G.: Long range scattering for the Maxwell–Schrödinger system with large magnetic field data and small Schrödinger data. Publ. Res. Inst. Math. Sci. 42(2), 421–459 (2006)
Ginibre, J., Velo, G.: Long range scattering and modified wave operators for the Maxwell–Schrödinger system. II. The general case. Ann. Henri Poincaré 8(5), 917–994 (2007)
Nakamura, M., Wada, T.: Local wellposedness for the Maxwell–Schrödinger equation. Math. Ann. 332(3), 565–604 (2005)
Nakamura, M., Wada, T.: Global existence and uniqueness of solutions to the Maxwell–Schrödinger equations. Commun. Math. Phys. 276(2), 315–339 (2007)
Ginibre, J., Velo, G.: Long range scattering for the Maxwell–Schrödinger system with arbitrarily large asymptotic data. Hokkaido Math. J. 37(4), 795–811 (2008)
Bejenaru, I., Tataru, D.: Global wellposedness in the energy space for the Maxwell–Schrödinger system. Commun. Math. Phys. 288(1), 145–198 (2009)
Wada, T.: Smoothing effects for Schrödinger equations with electro-magnetic potentials and applications to the Maxwell–Schrödinger equations. J. Funct. Anal. 263(1), 1–24 (2012)
Petersen, K.: Existence of a unique local solution to the many-body Maxwell–Schrödinger initial value problem. ArXiv e-prints (2014)
Antonelli, P., D’Amico, M., Marcati, P.: Nonlinear Maxwell–Schrödinger system and quantum magneto-hydrodynamics in 3D. Commun. Math. Sci. 15(2), 451–479 (2017)
Grafakos, L., Seungly, O.: The Kato-Ponce inequality. Commun. Partial Differ. Equ. 39(6), 1128–1157 (2014)
Wang, B.: Harmonic Analysis Methods for Nonlinear Evolution Equations, I. World Scientific Publishing Co., Singapore (2011)
Tao, T.: Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI. Local and global analysis (2006)
Strichartz, R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977)
Ginibre, J., Velo, G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133(1), 50–68 (1995)
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, New York (2010)
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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Communicated by R. Seiringer
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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