Rotating two-dimensional Bose--Einstein condensates in power-law traps: collapse, Thomas--Fermi scaling, and the inverse-square endpoint

Abstract

In two dimensions, the kinetic and contact energies scale identically under dilation, a trap $V=r^s/s$ as $λ^{-s}$. We follow this balance through the power-law family $-2\le s\le4$, $s\neq0$, for both signs of the coupling $g$, with rotation, using scaling bounds, radial solutions, Bogoliubov spectra, and two-dimensional imaginary-time states. Attraction collapses the cloud at the Townes coupling $g_{*}$ in every trap, with width $\ell\proptoδ^{1/(s+2)}$, $δ=1-|g|/g_{*}$; repulsion spreads it to a Thomas--Fermi (TF) radius $\propto g^{1/(s+2)}$. A rotating collapse leaves the axis once the rotation frequency $Ω$ exceeds its dipole frequency, which softens only for $s>2$; in a quartic trap this happens at $δ\simeq0.32\,Ω^6$, within $3\%$ of the Bogoliubov threshold; it then collapses on the minimum ring of the rotating-frame potential, which also empties a repulsive TF cloud at $Ω=(3g/π)^{1/6}$. For repulsive vortex states the TF energy error splits exactly into a density-gradient and a flow part of comparable or larger size. At $s=-2$ only the cutoff sets a length: the energy is exactly covariant with it, the condensate falls to the center for both signs, and repulsion leaves a halo of density $\propto1/(gr^2)$ whose extent grows exponentially with $g$. Near collapse, split-step imaginary time has a first-order stationary bias invisible to residual tests.