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A rigorous derivation of the ``miracle'' identity of three-dimensional inverse scattering
The large-energy asymptotic behavior of scattering solutions of the three-dimensional time-dependent Schrödinger equation is investigated. The second term of the expansion leads to the ``miracle'...
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Remarks on inverse scattering in one dimension

J. Math. Phys. 25, 2991 (1984); doi:10.1063/1.526014

Issue Date: October 1984

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Roger G. Newton
Physics Department, Indiana University, Bloomington, Indiana 47405
This paper answers the following questions: (1) what are the consequences in the matrix-Marchenko inversion scheme if a given S matrix lacks forward analyticity; and (2) in particular, does the condition known as the miracle depend on forward analyticity, and if not, what properties of S does it depend on? The answers are (1) if the input S matrix lacks forward analyticity then the output S matrix has it anyway, and (2) integrability of kRl,r is sufficient for the miracle to occur. It is also found that the matrix–Marchenko procedure simultaneously constructs the potentials for two scattering problems which differ only by the signs of their reflection coefficients. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
History: Received 23 November 1983; accepted 25 May 1984
Permalink: http://link.aip.org/link/?JMAPAQ/25/2991/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.Nk
    Classical and quantum physics: mechanics and fields Quantum theory; quantum mechanics Nonrelativistic scattering theory
  • 11.20.-e
    General theory of fields and particles S-matrix theory
  • YEAR: 1984

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
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REFERENCES (10)
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  1. I. Kay and H. E. Moses, Nuovo Cimento 3, 276 (1956).
  2. L. D. Faddeev, Trudy Mat. Inst. Stekl. 73, 314 (1964)
  3. [Am. Math. Soc. Transl. 2, 139 (1967)].
  4. P. Deift and E. Trubowitz, Commun. Pure Appl. Math. 32, 121 (1979).
  5. R. G. Newton, J. Math. Phys. 21, 493 (1980).
  6. An asterik denotes the complex conjugate, a tilde the transpose, and a dagger the Hermitian adjoint.
  7. P. B. Abraham, B. DeFacio, and H. E. Moses, Phys. Rev. Lett. 46, 1657 (1981);
    8. K. R. Brownstein, Phys. Rev. D 25, 2704 (1982).
    The potentials in these examples, however, are distributions.
  8. In three dimensions the miracle consists of a condition similar to (2.9), namely the theta-independence of Eq. (13) in R. G. Newton, Phys. Rev. Lett. 43, 541 (1979);
    9. or of Eq. (4.3) in R. G. Newton, J. Math. Phys. 21, 1698 (1980).
    See also Eq. (7.3) of C. S. Morawetz, Comp. Math. Appl. 7, 319 (1981).
    It was pointed out below Eq. (3.6) of R. G. Newton, J. Math. Phys. 23, 594 (1982),
    that the analog of Eq. (2.10) holds in that case, but in constrast to the one-dimensional case, this does not imply the miracle.
  9. N. I. Muskhelishvili, Singular Integral Equations (Noordhoff, Leyden, 1977).
  10. See Ref. 4 as corrected by R. G. Newton, J. Math. Phys. 24, 2152 (1983), footnote 20.
  11. See Ref. 8 or R. G. Newton, J. Math. Phys. 23, 2257 (1982), Lemma 2.

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