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arXiv · 1105.0512

Second order semiclassics with self-generated magnetic fields

Laszlo Erdos·Soren Fournais·Jan Philip Solovej
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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 $β\int B^2$ and we minimize over all magnetic fields. The parameter $β$ effectively determines the strength of the field. We consider the weak field regime with $βh^{2}\ge {const}>0$, 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 $h^{1+\e}$, 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 \cite{EFS3} to prove the second order Scott correction to the ground state energy of large atoms and molecules.

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Laszlo Erdos, Soren Fournais, Jan Philip Solovej. 2011-09-25. Second order semiclassics with self-generated magnetic fields. https://doi.org/10.1007/s00023-011-0150-z

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