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首页> 外文期刊>Journal of the European Mathematical Society: JEMS >Stability and semiclassics in self-generated fields
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Stability and semiclassics in self-generated fields

机译：自生领域的稳定性和半经典性

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We consider non-interacting particles subject to a fixed external potential V and a selfgenerated magnetic field B. The total energy includes the field energy β R B2 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 ! 0, of the total ground state energy E.β ; h; V /. The relevant parameter measuring the field strength in the semiclassical limit is ≤ D βh.We are not able to give the exact leading order semiclassical asymptotics uniformly in ≤ or even for fixed ≤. We do however give upper and lower bounds on E with almost matching dependence on ≤. In the simultaneous limit h ! 0 and ≤ ! 1 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 Schr?dinger operator.
机译：我们考虑非相互作用粒子受到固定的外部电势V和自生磁场B的影响。总能量包括场能βR B2，并且我们将所有粒子状态和磁场最小化。在spin-1 = 2粒子的情况下，这种最小化导致耦合的Maxwell-Pauli系统。参数β调整场与粒子之间的耦合强度，并有效地确定场的强度。我们研究稳定性和半经典渐近性h！总基态能量E.β的0; H; V /。在半经典极限中测量场强的相关参数为≤Dβh。我们无法均匀给出≤甚至对于固定≤的精确的前导半经典渐近性。但是，我们确实给出了E的上限和下限，并且对≤几乎具有依赖关系。在同时极限h！ 0且≤！从图1可以看出，标准的非磁性Weyl渐近线成立。对于无旋转情况，即Pauli运算符被Schr？dinger运算符代替的情况，同样的结果也成立。

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