Uniform Asymptotic Approximation Method with Pöschl–Teller Potential

In this paper, we study analytical approximate solutions for second-order homogeneous differential equations with the existence of only two turning points (but without poles) by using the uniform asymptotic approximation (UAA) method. To be more concrete, we consider the Pöschl–Teller (PT) potential...

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Bibliographic Details
Published inUniverse (Basel) Vol. 9; no. 11; p. 471
Main Authors Pan, Rui, Marchetta, John Joseph, Saeed, Jamal, Cleaver, Gerald, Li, Bao-Fei, Wang, Anzhong, Zhu, Tao
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.11.2023
Subjects
Approximation
Approximation method
black holes
Book publishing
cosmological perturbations
Differential equations
gravitational waves
Investigations
loop quantum cosmology
Methods
Ordinary differential equations
power spectrum
quasi-normal modes
Spacetime
United Kingdom
Germany
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Summary:In this paper, we study analytical approximate solutions for second-order homogeneous differential equations with the existence of only two turning points (but without poles) by using the uniform asymptotic approximation (UAA) method. To be more concrete, we consider the Pöschl–Teller (PT) potential, for which analytical solutions are known. Depending on the values of the parameters involved in the PT potential, we find that the upper bounds of the errors of the approximate solutions in general are ≲0.15∼10% for the first-order approximation of the UAA method. The approximations can be easily extended to high orders, for which the errors are expected to be much smaller. Such obtained analytical solutions can be used to study cosmological perturbations in the framework of quantum cosmology as well as quasi-normal modes of black holes.
ISSN:2218-1997
2218-1997
DOI:10.3390/universe9110471