Stellar rank under the contraction of SU(1,1) to the Heisenberg-Weyl group

Abstract

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 Poincar\'e 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.