For Ginzburg-Landau vortices of degree 2 and 3, the only bounded zero modes are the three geometric symmetries, and the Morse indices are 2 and 6.
Estimates for the Gross-Pitaevskii equation linearized around a vortex
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We consider the linearized two-dimensional Gross-Pitaevskii equation around a vortex of degree one, with data in the same equivariance class. Various estimates are proved for the solution; in particular, conditions for optimal decay in $L^\infty$ and boundedness in $L^2$ are identified. The analysis relies on a full description of the spectral resolution of the linearized operator through the associated distorted Fourier transform.
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Nondegeneracy and Morse Index of Ginzburg--Landau Vortices
For Ginzburg-Landau vortices of degree 2 and 3, the only bounded zero modes are the three geometric symmetries, and the Morse indices are 2 and 6.