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Scattering in one dimension: The coupled Schrödinger equation, threshold behaviour and Levinson's theorem

J. Math. Phys. 37, 6033 (1996); doi:10.1063/1.531762

Issue Date: December 1996

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K. A. Kiers
Department of Physics, University of British Columbia, Vancouver, British Columbia, V6T 1Z1 Canada

W. van Dijk
Redeemer College, Ancaster, Ontario, L9G 3N6 Canada
Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1 Canada
We formulate scattering in one dimension due to the coupled Schrödinger equation in terms of the S matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson's theorem is seen to have the form eta(0)=pi(nb+1/2n–1/2N), where eta(0) is the phase of the S matrix at zero energy, nb the number of bound states with nonzero binding energy, n the number of half-bound states, and N the number of coupled equations. In view of the effects due to the half-bound states, the threshold behaviour of the scattering amplitudes is investigated in general, and is also illustrated by means of particular potential models. ©1996 American Institute of Physics.
History: Received 28 May 1996; accepted 8 August 1996
Permalink: http://link.aip.org/link/?JMAPAQ/37/6033/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.Nk
    Classical and quantum physics: mechanics and fields Quantum mechanics Nonrelativistic scattering theory
  • 03.65.Nk
    Classical and quantum physics: mechanics and fields Quantum mechanics Nonrelativistic scattering theory
  • 02.30.Em
    Mathematical methods in physics Function theory, analysis Potential theory
  • 03.65.Ge
    Classical and quantum physics: mechanics and fields Quantum mechanics Solutions of wave equations: bound states
  • 11.55.-m
    General theory of fields and particles S-matrix theory; analytic structure of amplitudes
  • 02.10.-v
    Mathematical methods in physics Logic, set theory, and algebra
  • 11.80.Gw
    General theory of fields and particles Relativistic scattering theory Multichannel scattering
  • YEAR: 1996

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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