  We consider an inverse spectral theory in a domain with the cavity that is
bounded by a penetrable inhomogeneous medium. An ODE system is constructed
piecewise through the solutions inside and outside the cavity. The ODE system
is connected to the PDE system via the analytic continuation. For each
scattered angle, we describe its eigenvalue density in the complex plane, and
prove an inverse uniqueness on the inhomogeneity by the measurements in the
far-fields.
