  In this work we revisit axial perturbations of spherically symmetric and
non-rotating neutron stars. Although it has been object of many studies, it
still offers new insights that are of potential interest for more realistic
scenarios or in the study exotic compact objects, which have drawn much
attention recently. By using WKB theory, we first derive a new Bohr-Sommerfeld
rule that allows to investigate the quasi-normal mode spectrum and address the
inverse spectrum problem. The pure analytical treatment of the wave equation is
rather involved, because it requires the solution of the TOV equations and the
non-trivial tortoise coordinate transformation depending on the underlying
space-time. Therefore we provide an easy way to construct potentials that
simplifies the analytical treatment, but still captures the relevant physics.
The approximated potential can be used for calculations of the axial
perturbation spectrum. These results are also useful in the treatment of the
inverse problem. We demonstrate this by reconstructing the time-time component
of the metric throughout the star and constraining the equation of state in the
central region. Our method also provides an analytical explanation of the
empirically known asteroseismology relation that connects the fundamental QNM
and radius of a neutron star with its compactness.
