Inverse problem in energy-dependent potentials using semiclassical methods

• URL: http://arxiv.org/abs/2404.11478v2
• Date: Tue, 4 Jun 2024 22:55:23 GMT
• Title: Inverse problem in energy-dependent potentials using semiclassical methods
• Authors: Saulo Albuquerque, Sebastian H. Völkel, Kostas D. Kokkotas,
• Abstract summary: We use the semi-classical WKB method to first revisit the perturbation of bound states of potential wells and reflection/transmission coefficients. We then discuss the inverse problem, in which the latter observables are used to reconstruct the properties of the potentials.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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