Excited Proca stars are always unstable: under perturbations they collapse, disperse, or migrate to the ground state, with migration possible only for a narrow set of negative-binding-energy configurations.
The perturbation in φ (or F ) must be even with respect to reflections on the origin to maintain regularity there
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Proca stars in excited states
Excited Proca stars are always unstable: under perturbations they collapse, disperse, or migrate to the ground state, with migration possible only for a narrow set of negative-binding-energy configurations.