Second order semiclassics with self-generated magnetic fields
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Standard
Second order semiclassics with self-generated magnetic fields. / Erdös, Laszlo; Fournais, Søren; Solovej, Jan Philip.
I: Annales Henri Poincare, Bind 13, Nr. 4, 2012, s. 671-713.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
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 -
ID: 40301857