Stability of multi-state configurations of fuzzy dark matter

We study the stability properties of multi-state configurations of the Schr\"odinger-Poisson system without self-interaction, with monopolar and first dipolar components $(1,0,0)$+$(2,1,0)$. We show these configurations studied are stable using numerical simulations, and using criteria of stationarity, unitarity and time dependence consistency. The study covers a range of states with monopolar to dipolar mass ratio between 47 and 0.17. The astrophysical implication of this result is that this type of configurations is at least stable and can be considered physically sound in multi-state ultralight bosonic dark matter.