Second order semiclassics with self generated magnetic fields

L. Erdös, S. Fournais, J. Solovej, Annales Henri Poincare 13 (2012) 671–730.

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Abstract
We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field B. We also add the field energy β ∫ B 2 and we minimize over all magnetic fields. The parameter β effectively determines the strength of the field. We consider the weak field regime with βh 2 ≥ const > 0, where h is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor h 1+e{open}, i. e. the subleading term vanishes. However for potentials with a Coulomb singularity, the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper (Erdo{double acute}s et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011) to prove the second order Scott correction to the ground state energy of large atoms and molecules.
Publishing Year
Date Published
2012-05-01
Journal Title
Annales Henri Poincare
Volume
13
Issue
4
Page
671 - 730
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Erdös L, Fournais S, Solovej J. Second order semiclassics with self generated magnetic fields. Annales Henri Poincare. 2012;13(4):671-730. doi:10.1007/s00023-011-0150-z
Erdös, L., Fournais, S., & Solovej, J. (2012). Second order semiclassics with self generated magnetic fields. Annales Henri Poincare, 13(4), 671–730. https://doi.org/10.1007/s00023-011-0150-z
Erdös, László, Søren Fournais, and Jan Solovej. “Second Order Semiclassics with Self Generated Magnetic Fields.” Annales Henri Poincare 13, no. 4 (2012): 671–730. https://doi.org/10.1007/s00023-011-0150-z.
L. Erdös, S. Fournais, and J. Solovej, “Second order semiclassics with self generated magnetic fields,” Annales Henri Poincare, vol. 13, no. 4, pp. 671–730, 2012.
Erdös L, Fournais S, Solovej J. 2012. Second order semiclassics with self generated magnetic fields. Annales Henri Poincare. 13(4), 671–730.
Erdös, László, et al. “Second Order Semiclassics with Self Generated Magnetic Fields.” Annales Henri Poincare, vol. 13, no. 4, Birkhäuser, 2012, pp. 671–730, doi:10.1007/s00023-011-0150-z.

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