  In this work, we study the inverse problem of analog gravity systems which
admit rotation and energy-dependent boundary conditions. By extending two
recent results, we provide a recipe that allows one to relate resonant
transmission spectra with effective potentials and even reconstruct the
boundary condition at the core. Our methodology is based on the WKB method and
the identification of universal features in the transmission. One of the main
advantages of this method is that it is parameter-free, and relies only on
general properties of the underlying potential, instead of specific models.
While the reconstruction of underlying potentials is generally not uniquely
possible, the inverse method provides effective potentials with similar
spectral properties to the original one. To demonstrate the accuracy and scope
of our method, we apply it to a rotating imperfect draining vortex, which has
been proposed as an analog system to astrophysical extreme compact objects. We
conclude that the capability to explore energy-dependent boundary conditions
could be of interest for experimental studies of such systems.
