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Generalized approximate methods for transmission through a barrier governed by a differential equation of order 2n

Published online by Cambridge University Press:  24 October 2008

John Heading
John Heading
Affiliation:
University College of Wales, Aberystwyth
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Abstract

The potential barrier governed by a second-order ordinary differential equation has been studied for decades, both exactly and approximately. These concepts are generalized so as to be applicable to differential equations of arbitrary even order. The barrier is defined, and the approximate solutions both inside and outside are derived. Transmission through this barrier is investigated by deriving connexion formulae through the two transition points of arbitrary odd order. The transmitted solution is linked through the barrier to the incident and reflected solutions in a unique manner, enabling an approximate transmission coefficient to be calculated and its structure to be ascertained. Every concept in the investigation is a suitable generalization of the elementary case, the generalization possible being a product of the approximate processes adopted and how they limit the nature of the generalization. An inspection of the elementary case and its transmission coefficient gives no hint as to the nature of the generalization, the limitations necessarily imposed, and the structure of the final transmission coefficient, since the elementary case has special features not present in the generalization.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 2 , March 1979 , pp. 361 - 377
Copyright
Copyright © Cambridge Philosophical Society 1979

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