Stability and semiclassics in self-generated fields

L. Erdös, S. Fournais, J. Solovej, Journal of the European Mathematical Society 15 (2013) 2093–2113.


Journal Article | Published | English
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Abstract
We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B. The total energy includes the field energy β∫B2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, h→0, of the total ground state energy E(β,h,V). The relevant parameter measuring the field strength in the semiclassical limit is κ=βh. We are not able to give the exact leading order semiclassical asymptotics uniformly in κ or even for fixed κ. We do however give upper and lower bounds on E with almost matching dependence on κ. In the simultaneous limit h→0 and κ→∞ we show that the standard non-magnetic Weyl asymptotics holds. The same result also holds for the spinless case, i.e. where the Pauli operator is replaced by the Schrödinger operator.
Publishing Year
Date Published
2013-10-16
Journal Title
Journal of the European Mathematical Society
Volume
15
Issue
6
Page
2093 - 2113
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Cite this

Erdös L, Fournais S, Solovej J. Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. 2013;15(6):2093-2113. doi:10.4171/JEMS/416
Erdös, L., Fournais, S., & Solovej, J. (2013). Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society, 15(6), 2093–2113. https://doi.org/10.4171/JEMS/416
Erdös, László, Søren Fournais, and Jan Solovej. “Stability and Semiclassics in Self-Generated Fields.” Journal of the European Mathematical Society 15, no. 6 (2013): 2093–2113. https://doi.org/10.4171/JEMS/416.
L. Erdös, S. Fournais, and J. Solovej, “Stability and semiclassics in self-generated fields,” Journal of the European Mathematical Society, vol. 15, no. 6, pp. 2093–2113, 2013.
Erdös L, Fournais S, Solovej J. 2013. Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. 15(6), 2093–2113.
Erdös, László, et al. “Stability and Semiclassics in Self-Generated Fields.” Journal of the European Mathematical Society, vol. 15, no. 6, European Mathematical Society, 2013, pp. 2093–113, doi:10.4171/JEMS/416.

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