Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction

We perform accurate investigation of stability of localized vortices in an effectively two-dimensional ('' pancake-shaped '') trapped Bose-Einstein condensate with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity S = 1 are stable in a third of their existence region, 0 < N < (1/3)N-max((S=1)), where N is the number of atoms, and N-max((S=1)) is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, (1/3)N-max((S=1)) < N < 0.43N(max)((S=1)), the unstable vortex periodically splits in two fragments and recombines. At N > 0.43N(max)((S=1)), the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full three-dimensional (3D) Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio root 10, the stability interval of the S=1 vortices occupies approximate to 40% of their existence region, hence the two-dimensional (2D) limit provides for a reasonable approximation in this case. For the isotropic 3D configuration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to 2. All vortices with S >= 2 are unstable.