  In previous work we constructed new boson star solutions consisting of a
family of massive complex scalar fields minimally coupled to gravity in which
the individual fields have angular momentum, yet the configuration as a whole
is static and spherically symmetric. In the present article we study the linear
stability of these $\ell$-boson stars with respect to time-dependent, radial
perturbations. The pulsation equations, governing the dynamics of such
perturbations are derived, generalizing previous work initiated by M. Gleiser,
and shown to give rise to a two-channel Schr\"odinger operator. Using standard
tools from the literature, we show that for each fixed value $\ell$ of the
angular momentum number, there exists a family of $\ell$-boson stars which are
linearly stable with respect to radial fluctuations; in this case the
perturbations oscillate in time with given characteristic frequencies which are
computed and compared with the results from a nonlinear numerical simulation.
Further, there is also a family of $\ell$-boson stars which are linearly
unstable. The two families are separated by the configuration with maximum
mass. These results are qualitatively similar to the corresponding stability
results of the standard boson stars with $\ell=0$, and they imply the existence
of new stable configurations that are more massive and compact than usual boson
stars.
