  In this work we demonstrate how different semi-classical methods can be
combined in a novel way to reconstruct the perturbation potential of ultra
compact stars. Besides rather general assumptions, the only specific
information entering this approach is the spectrum of the $\textit{trapped}$
axial quasi-normal modes. In general it is not possible to find a unique
solution for the potential in the inverse problem, but instead a family of
potentials producing the same spectrum. Nevertheless, this already determines
important properties of the involved potential and can be used to rule out many
candidate models. A unique solution was found based on the additional natural
assumption that the exterior part ($r \gtrsim 3\,M$) is described by the
Regge-Wheeler potential. This is true in general relativity for any
non-rotating spherically symmetric object. This technique can be potentially
applied for the study of deviations from general relativity. The methods we
demonstrate are easy to implement and rather general, therefore we expect them
also to be interesting for other fields where inverse spectrum problems are
studied, e.g. quantum physics and molecular spectroscopy.
