AI 中文总结
本文研究 SU(1,1) 收缩到 Heisenberg-Weyl 群时 Husimi 函数零点的重分布，确定 stellar rank 的极限行为，证明 rank 上半连续、退化条件及保留最低 2r+3 个振幅的最优性。
AI 中文摘要
在 $SU(1,1)$ 到 Heisenberg-Weyl 群的收缩下，态的 Husimi 函数的零点按照单一尺度重新分布，我们确定了哪些零点得以保留。单模玻色子纯态的 stellar rank（即这些零点的数目）通过 rank 的消失来刻画平面上的高斯态。在庞加莱圆盘上，同样的构造适用于 $SU(1,1)$ 的离散序列，但情况并非如此，因为零点集留下了一个无零因子未被确定。因此，我们关注的是零点结构在收缩极限下的行为，而不是在固定的 Bargmann 指数下如何分类。两个希尔伯特空间之间的全部差异归结为一个权重序列 $w_k(m)=(2k)^m/(2k)_m$，其在 $k$ 上的单调性驱动了以下所有估计。重标度后的态构成一个正规族，每个非零极限点都位于 Segal-Bargmann 空间中，而那些具有有限个零点的态恰好是有限秩的 Fock 态。rank 是上半连续的，当且仅当所有零点按 $O(1/\sqrt{2k})$ 尺度缩放（即相干态标签的尺度）时取等号。一个双侧 Harnack 估计在紧集上确定了态的轮廓。另外，一个秩为 $r$ 的态族退化当且仅当最低的 $r+1$ 个权重向量上的振幅趋于零。质量反而在重标度圆盘的边缘逃逸，这是通过半经典 Husimi 势在一显式临界耦合处的一阶相变实现的。最后，收缩极限恰好保留最低的 $2r+3$ 个振幅，且这个数目不能进一步降低。
英文摘要
Under the contraction of $SU(1,1)$ to the Heisenberg-Weyl group, the zeros of the Husimi function of a state redistribute according to a single scale, and we determine which of them survive. The stellar rank of a single-mode bosonic pure state, the number of these zeros, characterises the Gaussian states on the plane through the vanishing of the rank. On the Poincaré disk, where the same construction applies to the discrete series of $SU(1,1)$, it does not, because the zero set leaves a zero-free factor undetermined. We therefore ask how the zero structure behaves in the contraction limit rather than how it should be classified at fixed Bargmann index. The entire difference between the two Hilbert spaces reduces to one weight sequence $w_k(m)=(2k)^m/(2k)_m$, whose monotonicity in $k$ drives every estimate below. The rescaled states form a normal family, every nonzero limit point lies in the Segal-Bargmann space, and those with finitely many zeros are exactly the finite-rank Fock states. The rank is upper semicontinuous, with equality precisely when all zeros scale as $O(1/\sqrt{2k})$, the scale of the coherent-state labels. A two-sided Harnack estimate pins the profile on compact sets. Separately, a family of rank $r$ degenerates if and only if the amplitudes on the lowest $r+1$ weight vectors tend to zero. Mass escapes at the edge of the rescaled disk instead, by a first-order transition of the semiclassical Husimi potential at an explicit critical coupling. Finally, the contraction limit retains exactly the lowest $2r+3$ amplitudes, and this number cannot be lowered.
Comments46 pages, 2 figures, 6 tables
