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On the Approximate Evaluation of Some Oscillatory Integrals

Institute of Physics, 10000 Zagreb, Croatia
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Atoms 2019, 7(2), 47; https://doi.org/10.3390/atoms7020047
Received: 4 March 2019 / Revised: 27 April 2019 / Accepted: 29 April 2019 / Published: 5 May 2019
(This article belongs to the Special Issue Atomic and Ionic Collisions with Formation of Quasimolecules)
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Abstract

To determine the photon emission or absorption probability for a diatomic system in the context of the semiclassical approximation it is necessary to calculate the characteristic canonical oscillatory integral which has one or more saddle points. Integrals like that appear in a whole range of physical problems, e.g., the atom–atom and atom–surface scattering and various optical phenomena. A uniform approximation of the integral, based on the stationary phase method is proposed, where the integral with several saddle points is replaced by a sum of integrals each having only one or at most two real saddle points and is easily soluble. In this way we formally reduce the codimension in canonical integrals of “elementary catastrophes” with codimensions greater than 1. The validity of the proposed method was tested on examples of integrals with three saddle points (“cusp” catastrophe) and four saddle points (“swallow-tail” catastrophe). View Full-Text
Keywords: oscillatory integrals; stationary point approximation; semi-classical theory; uniform Airy approximation oscillatory integrals; stationary point approximation; semi-classical theory; uniform Airy approximation
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Beuc, R.; Movre, M.; Horvatić, B. On the Approximate Evaluation of Some Oscillatory Integrals. Atoms 2019, 7, 47.

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