Second order semiclassics with self-generated magnetic fields
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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 $\beta \int B^2$ and we minimize over all magnetic fields. The parameter $\beta$ effectively determines the strength of the field. We consider the weak field regime with $\beta h^{2}\ge {const}>0$, 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.
Original language | English |
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Journal | Annales Henri Poincare |
Volume | 13 |
Issue number | 4 |
Pages (from-to) | 671-713 |
Number of pages | 43 |
ISSN | 1424-0637 |
DOIs | |
Publication status | Published - 2012 |
Links
- http://arxiv.org/abs/1105.0512
Accepted author manuscript
ID: 40301857