自生光场中的稳定性与半经典
Stability and semiclassics in self-generated fields
摘要：
我们考虑非相互作用的粒子受到一个固定的外部势$V$和一个自生磁场$B$。总能量包括场能量$β\int B^2$，并且我们在所有粒子状态和磁场上最小化。在自旋为1/2的粒子的情况下，这种最小化导致耦合麦克斯韦-泡利系统。参数$β$调节场与粒子之间的耦合强度，它有效地决定了场的强度。我们研究了总基态能量$E(β, h, V)$的稳定性和半经典渐进性$h\to0$。测量半经典极限中的场强的相关参数是$κ=βh$。我们不能在$κ$中甚至对固定的$κ$给出精确的首阶半经典渐近性。然而，我们给$E$的上限和下限几乎匹配依赖于$κ$。在同时限制$h\to0$和$κ\to\infty$，我们表明，标准的非磁性外尔渐近举行。同样的结果也适用于无自旋的情况，即泡利算符被薛定谔算符取代。
英文摘要：
We consider non-interacting particles subject to a fixed external potential $V$ and a self-generated magnetic field $B$. The total energy includes the field energy $β\int B^2$ and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter $β$ tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, $h\to0$, of the total ground state energy $E(β, h, V)$. The relevant parameter measuring the field strength in the semiclassical limit is $κ=βh$. We are not able to give the exact leading order semiclassical asymptotics uniformly in $κ$ or even for fixed $κ$. We do however give upper and lower bounds on $E$ with almost matching dependence on $κ$. In the simultaneous limit $h\to0$ and $κ\to\infty$ we show that the standard non-magnetic Weyl asymptotics holds. The same result also holds for the spinless case, i.e. where the Pauli operator is replaced by the Schrödinger operator.
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