Abstract
This pedagogical survey is presented following a suggestion of the Workshop Committee. It is tried to present in a simple way the main ill-posed aspects of inverse problems arising in nuclear scattering. One also explains on some examples how they are dealt with or how they are circumvented.
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References
For more details, and references, see K. Chadan & P.C. Sabatier “Inverse Problems in Quantum Scattering Theory” Springer-Verlag, New York Heidelberg Berlin 1977.
Ref (1) page 146 and ff.
Ref (1) page 64
For recent references on the three-dimensional inverse problem see R.G. Newton “The Marchenko and Gel'fand Levitan Methods in the Inverse Scattering Problem in one and three dimensions” in “Conference on Inverse Scattering: Theory and Application” J. Bee Bednar et al. ed. SIAM Philadelphia (1983) and (same author)
A Faddeev — Marchenko method for Inverse Scattering in three dimensions. Inverse Problems 2 (1985). A study of the ill-posed aspects of the inverse problem at fixed energy in the class of truncated potentials by Y. Loubatières will be published soon.
P.C. Sabatier: Well-posed Questions and Exploration of the Space of Parameters in Linear and Non Linear Inversion. In “Inverse Problems of Acoustic and Elastic Waves” F. Santosa et al. Ed. SIAM Philadelphia 1984.
V. Glaser and A. Martin, H. Grosse, W. Thirring: A family of optimal conditions for the absence of bound states in a potential. In Studies in mathematical Physics (eds. E.H. Lieb, B. Simon, A.S. Wightman) Princeton U.P. 1976. See also B. Simon same ref. also: A. Martin and P.C. Sabatier: Impedance, zero energy wave function, and bound states. J. Math. Phys. 18, 1623–1626 (1977).
See for example P.E. Hodgson: Nuclear Reactions and Nuclear Structure — Clarenton Press, Oxford, 1971. For a recent monograph on applications of inversion theory, particulary in nuclear scattering, see Zachariev B.N. and Suzko A.A. “Potentials and quantum scattering direct and inverse problems”, to be published by Energoatomisdat (Moscow) in 1985).
A recent example is given by A.A. Ioannides and R.S. Mackintosh: A method for S-matrix to Potential Inversion at Fixed Energy. Nucl. Phys. A 438, 354 (1985).
For a review and references see: P.C. Sabatier: “Rational Reflection Coefficients in One-Dimensional Inverse Scattering and Applications. In “Conference on Inverse Scattering: Theory and Application” J. Bee Bednar et al. eds. SIAM, Philadelphia 1983.
R. Lipperheide and H. Fiedeldey: Inverse Problem for Potential Scattering at fixed Energy I: Z. Phys. A 286, 45–46 (1978) and id. II: Z. Phys. A 301, 81–89 (1981). R. Lipperheide, S. Sofianos and H. Fiedeldey Potential Inversion for scattering at fixed energy: Phys. Rev. C 26, 770–772 (1982).
H. Burger, L.J. Allen, H. Fiedeldy, S.A. Sofianos and R. Lipperheide: Potentials obtained by inversion of e — He atomic scattering data: Physics Lett. 97 A, 39–41 (1983). B.V. Rudyak, A.a; Surko and B.N. Zachariev: Exactly solvable models (Crum-Krein Transformation in the λ2, E, plane). Phys. Scripts 29, 515–517, (1984).
A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi: Higher transcendental functions. Mac Graw Hill Ed. 1953.
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Sabatier, P.C. (1985). Introduction to ill-posed aspects of nuclear scattering. In: Krappe, H.J., Lipperheide, R. (eds) Advanced Methods in the Evaluation of Nuclear Scattering Data. Lecture Notes in Physics, vol 236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-15990-8_1
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