AI 中文总结
本文研究二维环面上磁薛定谔流的长时间半经典测度，在几何非消失条件下证明量子唯一遍历性的动力学形式，并给出构型与动量边际的收敛阈值。
AI 中文摘要
我们研究了平坦二维环面上与磁薛定谔流相关的含时半经典测度在长时间范围内的行为。在有效磁力满足几何非消失假设的条件下，我们确定了两个阈值，并证明了此类系统具有量子唯一遍历性的动力学形式。若时间尺度 $\tau_h \gg h^{-1/2}$，则构型边际分布为 $\mathbb T^2$ 上的归一化勒贝格测度。此外，若 $\tau_h \gg h^{-1}$，则动量边际分布在每个正则水平集上条件于该曲线上的归一化不变测度。
英文摘要
We investigate time-dependent semiclassical measures associated with the magnetic Schrödinger flow on the flat two-dimensional torus in long-time regimes. Under a geometric nonvanishing assumption on an effective magnetic force, we identify two thresholds and prove a dynamical form of quantum unique ergodicity for such systems. If the time scale $τ_h \gg h^{-1/2}$, then the configuration marginal is the normalized Lebesgue measure on $\mathbb T^2$. In addition, if $τ_h \gg h^{-1}$, then the momentum marginal is, conditionally on each regular level set, the normalized invariant measure on that curve.
