In previous work we analyzed the linear stability of non-relativistic
ℓ-boson stars with respect to radial modes and showed that ground state
configurations are stable with respect to these modes, whereas excited states
are unstable. In this work we extend the analysis to non-spherical linear mode
perturbations. To this purpose, we expand the wave function in terms of tensor
spherical harmonics which allows us to decouple the perturbation equations into
a family of radial problems. By using a combination of analytic and numerical
methods, we show that ground state configurations with ℓ>1 possess
exponentially in time growing non-radial modes, whereas only oscillating modes
are found for ℓ=0 and ℓ=1. This leads us to conjecture that
nonrelativistic ℓ-boson stars in their ground state are stable for
ℓ=1 as well as ℓ=0, while ground state and excited configurations
with ℓ>1 are unstable.Comment: 21 pages, 5 figures, 2 table