  In this work we present a semi-classical approach to solve the inverse
spectrum problem for one-dimensional wave equations for a specific class of
potentials that admits quasi-stationary states. We show how inverse methods for
potential wells and potential barriers can be generalized to reconstruct
significant parts for the combined potentials. For the reconstruction one
assumes the knowledge of the complex valued spectrum and uses the exponential
smallness of its imaginary part. Analytic spectra are studied and a recent
application of the method in the literature for gravitational wave physics is
discussed. The method allows for a simple reconstruction of quasi-stationary
state potentials from a given spectrum. Thus it might be interesting for
different branches of physics and related fields.

Quantum Physics;General Relativity and Quantum Cosmology