WKB approximation for quasi-bound states and trapped modes
Abstract
We develop a largely automatic semi-analytic method for calculating weakly damped quasi-bound states and trapped modes supported by a local minimum of an effective potential. The real part of the frequency is obtained from an arbitrarily high-order local WKB expansion and Padé resummation, while the exponentially small imaginary part is estimated using the Gamow approximation for tunnelling through one or two potential barriers. Because the local quantization requires only derivatives of the potential at its minimum, the method applies readily to non-rational effective potentials and to different compact-object geometries. We provide a Mathematica notebook implementing the procedure. Comparisons with continued-fraction and direct numerical results for massive scalar fields around Schwarzschild and Kerr black holes and for axial trapped modes of a uniform-density star show that the method yields accurate real frequencies and useful estimates of decay rates outside the superradiant regime.
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BibTeX
@misc{konoplya2026approximation,
title = {{WKB approximation for quasi-bound states and trapped modes}},
author = {R. A. Konoplya and Z. Stuchlík and A. Zhidenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10880},
eprint = {2609.10880},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10880},
}
RIS
TY - UNPB
TI - WKB approximation for quasi-bound states and trapped modes
AU - R. A. Konoplya
AU - Z. Stuchlík
AU - A. Zhidenko
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.10880
UR - https://arxiv.org/abs/2609.10880
AB - We develop a largely automatic semi-analytic method for calculating weakly damped quasi-bound states and trapped modes supported by a local minimum of an effective potential. The real part of the frequency is obtained from an arbitrarily high-order local WKB expansion and Padé resummation, while the exponentially small imaginary part is estimated using the Gamow approximation for tunnelling through one or two potential barriers. Because the local quantization requires only derivatives of the potential at its minimum, the method applies readily to non-rational effective potentials and to different compact-object geometries. We provide a Mathematica notebook implementing the procedure. Comparisons with continued-fraction and direct numerical results for massive scalar fields around Schwarzschild and Kerr black holes and for axial trapped modes of a uniform-density star show that the method yields accurate real frequencies and useful estimates of decay rates outside the superradiant regime.
ER -