Abstract
The talk consists of two separate parts. In the first part we show that the multiple-scattering formulation of the quantum mechanical three body problem requires as input the half-off-shell two-body scattering amplitude. We indicate how this input information can be obtained from the physical (on-shell) two-body scattering amplitude using the methods of the inverse problem of scattering.
In the second part we present a non-conventional (dispersion theory) derivation of the basic equations of the inverse problem of scattering (for fixed 1); we obtain in addition to the well-known Marchenco and Gel'fand Levitan equations an infinite set of alternative equations.
We finally present a momentum space formulation of the inverse problem which is particularly suited for the application mentionned in the first part of the talk.
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References
The literature on the inverse problem of scattering is extensive and we do not intend to give references to the original literature here, with a few excep tions. General references are
L.D. FADDEEV, Usp. Mat. Nauk 14, 57 (1959) (Translation J. Math. Phys. 4, 72 (1963))
V. De ALFARO and T. REGGE, Potential Scattering (North-Holland, Amsterdam, 1965), Chap. 12.
R.G. NEWTON, Scattering Theory of Particles and Waves, (Mc. Graw-Hill, New-York, 1966)
The unitarityconstraints discussed in section 3 were first investigated by
M. BARANGER, B. GIRAUD, S.K. MUKHOPADRYAY and P.U. SAUER, Nucl. Phys. A138, 1 (1969)
A survey of the inverse problem for separable potentials is given in
K. CHADAN, in Mathematics of profile inversion, (NASA-Ames Research Center, Moffett Field, Calif. July 12–16, 1971)
The equation (4.17) and the momentum space methods in section 5 appeared in
B.R. KARLSSON, Phys. Rev. D 10, 1985 (1974).
Note that Eqns] (6.16) and (6.18) of this reference are incorrect, and should be replaced by Eqn (5–19) of this paper.
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© 1978 Springer-Verlag
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Karlsson, B. (1978). Inverse method for off-shell continuation of the scattering amplitude in quantum mechanics. In: Sabatier, P.C. (eds) Applied Inverse Problems. Lecture Notes in Physics, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09094-0_82
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DOI: https://doi.org/10.1007/3-540-09094-0_82
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