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Abstract

Wave equations with energy-dependent potentials appear in many areas of
physics, ranging from nuclear physics to black hole perturbation theory. In
this work, we use the semi-classical WKB method to first revisit the
computation of bound states of potential wells and reflection/transmission
coefficients in terms of the Bohr-Sommerfeld rule and the Gamow formula. We
then discuss the inverse problem, in which the latter observables are used as a
starting point to reconstruct the properties of the potentials. By extending
known inversion techniques to energy-dependent potentials, we demonstrate that
so-called width-equivalent or WKB-equivalent potentials are not isospectral
anymore. Instead, we explicitly demonstrate that constructing quasi-isospectral
potentials with the inverse techniques is still possible. Those reconstructed,
energy-independent potentials share key properties with the width-equivalent
potentials. We report that including energy-dependent terms allows for a rich
phenomenology, particularly for the energy-independent equivalent potentials.