TY - JOUR

T1 - Second order semiclassics with self-generated magnetic fields

AU - Erdös, Laszlo

AU - Fournais, Søren

AU - Solovej, Jan Philip

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

U2 - 10.1007/s00023-011-0150-z

DO - 10.1007/s00023-011-0150-z

M3 - Journal article

VL - 13

SP - 671

EP - 713

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 4

ER -