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 ∫ 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 ≥ 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{open}), 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 (Erdo{double acute}s et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011) to prove the second order Scott correction to the ground state energy of large atoms and molecules.
展开▼
机译：我们考虑具有外部电势和自生磁场B的三维Pauli算子的负特征值之和的半经典渐近性。我们还加上了磁场能量β∫B〜2，并且在所有磁场中都最小化。参数β有效地确定了场的强度。我们考虑βh〜2≥const> 0的弱场状态，其中h是半经典参数。对于平滑势，我们证明了总能量的半经典渐近性是由非磁性Weyl项以超前阶给出的，误差范围小于因子h〜（1 + e {open}），即i。 e。次要术语消失了。但是，对于具有库仑奇点的电势，由于奇点的非半经典效应，次导项不会消失。结合多尺度技术，这种精确的估计值可用于伴随论文（Erdo {double急性} s等人，《具有自生磁场的大分子的斯科特校正》，预印本，2011年）证明了二阶斯科特校正到大原子和分子的基态能量。
展开▼
