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In this paper we consider the generalized inverse iteration for computingground states of the Gross-Pitaevskii eigenvector problem (GPE). For that weprove explicit linear convergence rates that depend on the maximum eigenvaluein magnitude of a weighted linear eigenvalue problem. Furthermore, we show thatthis eigenvalue can be bounded by the first spectral gap of a linearizedGross-Pitaevskii operator, recovering the same rates as for linear eigenvectorproblems. With this we establish the first local convergence result for thebasic inverse iteration for the GPE without damping. We also show how ourfindings directly generalize to extended inverse iterations, such as theGradient Flow Discrete Normalized (GFDN) proposed in [W. Bao, Q. Du, SIAM J.Sci. Comput., 25 (2004)] or the damped inverse iteration suggested in [P.Henning, D. Peterseim, SIAM J. Numer. Anal., 53 (2020)]. Our analysis alsoreveals why the inverse iteration for the GPE does not react favourably tospectral shifts. This empirical observation can now be explained with a blow-upof a weighting function that crucially contributes to the convergence rates.Our findings are illustrated by numerical experiments.

The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem