Inverse problem in energy-dependent potentials using semiclassical methods

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.

12 pages, 11 figures

References in corpus (15)

• Quasinormal modes of black holes and black branes
• Quasi-Normal Modes of Stars and Black Holes
• Quasinormal modes of black holes: from astrophysics to string theory
• Analogue Gravity
• Quasinormal behavior of the D-dimensional Schwarzshild black hole and higher order WKB approach
• Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations
• Wave equations with energy dependent potentials
• Wormhole Potentials and Throats from Quasi-Normal Modes
• Gravitational wave sources: reflections and echoes
• Ultra Compact Stars: Reconstructing the Perturbation Potential
• The Inverse Problem for Hawking Radiation
• On the Inverse Spectrum Problem of Neutron Stars
• Inversion of Gamow's Formula and Inverse Scattering
• Inverse spectrum problem for quasi-stationary states
• Inverse problem of analog gravity systems

Cited by in corpus (3)

• Perturbing the vortex: quasinormal and quasibound spectra of rotating acoustic geometries
• Inverse problem of analog gravity systems II: Rotation and energy-dependent boundary conditions
• Bayesian analysis of analog gravity systems with the Rezzolla-Zhidenko metric