  The Bogoliubov equations are solved for a three-dimensional Bose-Einstein
condensate containing a doubly quantized vortex, trapped in a harmonic
potential. Complex frequencies, signifying dynamical instability, are found for
certain ranges of parameter values. The existence of alternating windows of
stability and instability, respectively, is explained qualitatively and
quantitatively using variational calculus and direct numerical solution. It is
seen that the windows of stability are much smaller for a cigar shaped
condensate than for a pancake shaped one, which is consistent with the findings
of recent experiments.
