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Inverse problem in energy-dependent potentials using semiclassical methods

Saulo Albuquerque1,2,*, Sebastian H. Völkel3,†, and Kostas D. Kokkotas2

  • 1Departmento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa 58059-900, Paraíba, Brazil
  • 2Theoretical Astrophysics, IAAT, University of Tübingen, D-72076 Tübingen, Germany
  • 3Max Planck Institute for Gravitational Physics (Albert Einstein Institute), D-14476 Potsdam, Germany

  • *saulo.filho@academico.ufpb.br
  • †sebastian.voelkel@aei.mpg.de
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Phys. Rev. D 109, 096014 – Published 13 May, 2024

DOI: https://doi.org/10.1103/PhysRevD.109.096014

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 semiclassical Wentzel-Kramers-Brillouin (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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Phys. Rev. D 109, 096014– Published 13 May, 2024
  • Received 25 February 2024
  • Accepted 16 April 2024
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