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2.—Barriers bounded by Two Transition Points of Arbitrary Odd Order

Published online by Cambridge University Press:  14 February 2012

John Heading
John Heading
Affiliation:
Department of Applied Mathematics, University College of Wales, Aberystwyth
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Synopsis

A generalisation is considered of the potential barrier problem beyond the familiar case in which the barrier is bounded by transition points of order one. Here, the two transition points involved are of arbitrary odd order. The approximate method employed, though formal in character, avoids certain pitfalls often made in the past whereby certain exponentially small terms within the barrier are confounded with inherent error terms. This confusion is avoided in the treatment given here by tracing uniformly approximate solutions round the transition points in the complex plane by means of the Stokes phenomenon, the method not requiring the dubious concept of a subdominant term existing in the presence of a dominant term on a Stokes line. At the same time, the solutions to which the reflection and transmission coefficients may be attached are carefully discussed, so that the appearance of a small exponential term may be seen to be genuine when taken in conjunction with inherent error terms. The resulting formula for the modulus of the reflection coefficient generalizes the more elementary formula.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 51 - 64
Copyright
Copyright © Royal Society of Edinburgh 1975

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References

1Budden, K. G., Radio Waves in the ionosphere. C.U.P., 1961.Google Scholar
2Fröman, N. and Fröman, P.O., JWKB approximation, contributions to the theory. Amsterdam: North-Holland, 1965.Google Scholar
3Heading, J., Proc. Cambridge Philos. Soc, 53, 399–418, 1957.CrossRefGoogle Scholar
4Heading, J., Proc. Cambridge Philos. Soc, 53, 419–441, 1957.CrossRefGoogle Scholar
5Heading, J., An introduction to phase-integral methods. London: Methuen, 1962.Google Scholar
6Heading, J., Proc. Cambridge Philos. Soc, 63, 439–450, 1967.CrossRefGoogle Scholar
7Heading, J., Proc. Cambridge Philos. Soc, 71, 353–367, 1972.CrossRefGoogle Scholar
8Heading, J., Physica 77, 263–278, 1974.CrossRefGoogle Scholar
9Jeffreys, H., Asymptotic approximations. Oxford: Clarendon, 1962.Google Scholar
10Kemble, E. C, The fundamental principles of quantum mechanics. New York: McGraw-Hill, 1937.Google Scholar
11Olver, F. W., J.Res. Nat. Bur. Standards Sect. B, 69, 291–300, 1965.CrossRefGoogle Scholar