Fake Airy functions and the asymptotics of reflectionlessness
Fake Airy functions and the asymptotics of reflectionlessness
Two classes of analytic refractive-index profile P2(z,ε ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ε ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P2 (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection.
Airy function, asymptotics, reflectionlessness, quantum mechanics, potential, quantum potential, reflection, reflection coefficient
L243-L246
Berry, Michael
ec39b1ad-7f54-4abf-9fcf-e5a3d1c2ab84
Howls, Christopher
66d3f0f0-376c-4f7a-a206-093935e6c560
1990
Berry, Michael
ec39b1ad-7f54-4abf-9fcf-e5a3d1c2ab84
Howls, Christopher
66d3f0f0-376c-4f7a-a206-093935e6c560
Berry, Michael and Howls, Christopher
(1990)
Fake Airy functions and the asymptotics of reflectionlessness.
Journal of Physics A: Mathematical and General, 23 (6), .
(doi:10.1088/0305-4470/23/6/002).
Abstract
Two classes of analytic refractive-index profile P2(z,ε ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ε ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P2 (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection.
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Published date: 1990
Keywords:
Airy function, asymptotics, reflectionlessness, quantum mechanics, potential, quantum potential, reflection, reflection coefficient
Identifiers
Local EPrints ID: 496795
URI: http://eprints.soton.ac.uk/id/eprint/496795
ISSN: 0305-4470
PURE UUID: 432d81a4-1c90-4acf-8631-0fcfed7e5166
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Date deposited: 08 Jan 2025 07:07
Last modified: 23 Aug 2025 01:44
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Author:
Michael Berry
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