Abstract
In the usual scattering-theory problem the hamiltonian of the system or the interparticle forces are known and a cross section, polarization, etc., are to be calculated and subsequently confronted with experimental results. The “inverse” problem is posed in the opposite direction: given certain kinds of information obtained more or less directly from scattering experiments, we are to determine the interparticle forces. Or before this problem can be solved: Is the given amount of information sufficient to determine these forces uniquely? If not, what kinds of ambiquity are there?
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Notes and References
For a discussion of the ambiguities of inferring the scattering amplitudes from cross section measurements of particles with spin, see R. van Wageningen (1965); M. J. Moravcsik (1968); Moravcsik and Yu (1969). The inverse scattering problem is treated in the WKB approximation in the following works: F. C. Hoyt (1939); O. B. Firsov (1953); J. W. Brackett et al. (1963); Sanders and Mueller (1963); P. C. Sabatier (1965 and 1973); Bernstein and O’Brien (1967); M. V. Berry (1969); Buck and Pauly (1969); W. H. Miller (1969 and 1971); G. Vollmer (1969); J. F. Boyle (1971); U. Buck (1971, 1972, 1974); Yu. N. Demkov et al. (1971); S. V. Khudyakov (1971); V. G. Rich et al. (1971); R. Klingbeil (1972); D. E. Pritchard (1972); Bernstein and LaBudde (1973); E. Kujawski (1973): M. Cuer (1977); Vasilevskii and Zhirnov (1978); R. B. Gerber et al. (1978); Chan and Lu (1980); L. Hüwel et al. (1981). The inverse problem for separable potentials (or sums of such) was solved by Gourdin and Martin (1957 and 1958); K. Chadan (1958 and 1967); Bolsterli and McKenzie (1965); Mills and Reading (1969); H. Fiedeldey (1969); F. Tabakin (1969); M. G. Fuda (1981). See also Garcilazo and Wilde (1980). An excellent general reference on the inverse scattering problem is Chadan and Sabatier (1977).
The first explicit example of a phase-shift ambiguity for spinless particles was given by J. H. Crichton (1966). The idea of using the generalized optical theorem as a means for determining the phase of the amplitude goes back to N. P. Klepikov (1962 and 1964) and, independently, to R. G. Newton (1968) and A. Martin (1969). For further analyses and applications, see L. M. Lazarev (1967): A. Gersten (1969); Gerber and Karplus (1970); H. Goldberg (1970); R. F. Alvarez-Estrada (1971); C. Eftimiu (1971, 1972, 1973); D. Atkinson et al. (1972); H. Burkhardt (1972); M. Tortorella (1972, 1974, 1975); Cornille and Martin (1972b); R. B. Gerber (1972); Alvarez-Estrada and Carreras (1973); Atkinson, Johnson, Mehta, and De Roo (1973); Atkinson, Johnson, and Warnock (1973); Atkinson, Mahoux, and Yndurain (1973 and 1975); G. R. Bart et al. (1973); Behrends and Ruijsenaars (1973); D. Atkinson et al. (1974); Boyce and Roberts (1974); Cornille and Drouffe (1974); M. DeRoo (1974); Atkinson, Kaebebeke, and DeRoo (1975); Tortorella and Leise (1976); D. Atkinson et al. (1977); 1. A. Sakmar (1978, 1979, 1980); R. B. Gerber et al. (1980). For additional use of analyticity to resolve ambiguities, see Itzykson and Martin (1973); G. R. Bart et al. (1974); Burkhardt and Martin (1975). See also Bessis and Martin (1967).
The first attempts at a solution of the inversion problem of this section were made by C. E. Fröberg (1947, 1948, 1949, and 1951); E. A. Hylleraas (1948). It was then shown by V. Bargmann (1949a and b) that the phase shift alone does not necessarily uniquely determine the potential. The uniqueness question was resolved by N. Levinson (1949a and b) and independently by V. A. Marchenko (1950) and G. Borg (1949) [see also (1946)]. The previous work of Fröberg and Hylleraas was amended in the light of the new knowledge by B. Holmberg (1952). New construction procedures were then invented by Jost and Kohn (1952a and b). Meanwhile the Gel’fand-Levitan equation had been introduced into the inverse Sturm-Liouville problem by Gel’fand and Levitan (1951a and b). It was applied to the inverse scattering problem by Jost and Kohn (1953) and by N. Levinson (1953). Since that time the problem has given rise to a rather extensive Russian mathematical literature, the main items in which are the following: L. A. Chudov (1949 and 1956); V. A. Marchenko (1952, 1953, 1955, 1960); Ju. M. Berezanskii (1953, 1955, 1958); A. Sh. Block (1953); M. G. Krein (1953a, b, c, 1954, 1955, 1956, 1957, 1958); M. G. Neuhaus (1955); B. Ya. Levin (1956); B. M. Levitan (1956); L. D. Faddeev (1956 and 1958); Agranovich and Marchenko (1957a and b, 1958); Gohberg and Krein (1958). See also P. Jauho (1950); K. Chadan (1955 and 1956); M. M. Crum (1955); T. Ohmura (1956); R. Jost (1956); R. G. Newton (1956); B. Friedman (1957); M. Petras (1962); M. Blazek (1962, 1963, 1964); E. A. Hylleraas (1964); W. A. Pearce (1964); J. Lanik (1964); A. G. Ramm (1965b); Swan and Pearce (1966); Benn and Scharf (1969); J. Weiss (1969); H. Cornille (1970a); J. Underhill (1970); Gugushvili and Mentkovsky (1972); Coz and Coudray (1973); F. Lambert et al. (1975); V. V. Malyarov et al. (1975a); H. Cornille (1976); R. J. W. Hodgson (1978); J. Wiesner et al. (1978); H. Moses (1979b); Brander and Chadan (1981); M. Coz (1981). An attempt at summing the contributions from all/was made by M. Blazek (1966). P. B. Abraham et al. (1981) showed explicitly that in a wider class of potentials the inversion, even in the absence of bound states, is not unique. See also P. B. Abraham et al. (1982) for ambiguity if nonlocal potentials are admitted. Generalizations of the Jost-Kohn (or Gel’fand-Levitan and Marchenko) procedures to coupled equations were given by Newton and Jost (1955); M. G. Krein (1956); to the scattering of spin 1/2 particles with the tensor force present by R. G. Newton (1955); Agranovich and Marchenko (1958); [see also sec. 9 of the review by R. G. Newton (1960); V. E. Troitskii (1978)]; to the Klein-Gordon equation by E. Corinaldesi (1954a); A. Degasperis (1970); H. Cornille (1970b); J. Weiss (1971); Weiss and Scharf (1971); J. Jp. Leon (1980); to the Dirac Equation by Prats and Toll (1959); M. G. Gasimov (1966 and 1968); O. D. Corbella (1970); R. Weiss et al. (1972); to both relativistic equations by M.Verde (1958–1959); [see also S.N.Sokolov(1979); J. Jp. Leon (1981)]; to coupled channels by J. R. Cox (1962, 1966, 1967, 1975); M. Coz (1966); Cox and Garcia (1975); Coz and Rochus (1977); P. Rochus (1979); to energy dependent potentials by V. I. Mal’cenko (1966); M. Jaulent (1972 and 1975); Jaulent and Jean (1972, 1975, 1982); M. Tsutsumi (1981); to complex potentials by M. Jaulent (1976); De Facio and Moses (1980); to include Coulomb potentials by P. Swan (1967b); to singular potentials by Baeteman and Chadan (1975); K. Chadan (1978). For treatments of the inverse scattering problem in the momentum representation, see M. I. Sobel (1968); B. R. Karlsson (1972, 1974, 1978); for a generalization to other boundary conditions, see H. E. Moses (1978). Eq. (20.21) was first proved, in the context of the Marchenko equation, by H. Cornille (1967). For the solution of the inverse scattering problems in one dimension (from — oc to + oo) see A. Sh. Bloch (1953); I. Kay (1955); Kay and Moses (1956, 1957); H. E. Moses (1956); L. D. Faddeev (1958 and 1964); J. C. Portinari (1965); Deift and Trubowitz (1979); R. G. Newton (1980c, 1981a, appendix); see also M. G. Krein (1953a). For discussions of the inverse scattering problem in electromagnetic theory, see R. Mireles (1966); Wolf and Shewell (1967); Shewell and Wolf (1968); E. Wolf (1969); W. H. Carter (20.3). Other useful general references are the review by L. D. Faddeev (1959) and the monographs Agranovitch and Marchenko (1963), and Chadan and Sabatier (1977).
The content of this section follows R. G. Newton (1956). See also N. Levinson (1949). Equation (20.40) is a special case of the information contained in the momentsof the phase shift. For more general formulas of this nature see Buslaev and Faddeev (1960); V. S. Buslaev (1962, 1967, 1971); I. C. Percival (1962 and 1963); Percival and Roberts (1963); M. J. Roberts (1963,1964,1965); K. Chadan (1965 and 1966);Calogero and Degasperis (1968); Chadan and Montes (1968); F. Calogero (1971b); O. D. Corbella (1971); Kirhznitz and Takibaev (1972); R. D. Puff (1975); N. Zh. Takibaev (1977); D. Bollé (1979a); Bollé and Osborn (1980); see also F. Calogero et al. (1968).
The problem of obtaining the potential from a knowledge of all phase shifts at one energy was first approached by J. A. Wheeler (1955) from the point of view of the WKB approximation. Then T. Regge (1959) dealt with it by extending the angular momentum into the complex plane. The approach of Martin and Targonski (1961) is applicable to superpositions of Yukawa potentials only. The first general solution of the problem was given by R. G. Newton (1962a). The present section follows that paper for the most part. A mathematically amusing adjunct is provided by P. J. Redmond (1964). This work was carried further by P. C. Sabatier (1966, 1967, 1972b) [see also corrections on p. 337 of Chadan and Sabatier (1977)]; R. G. Newton (1967b); Cox and Thompson (1969 and 1970); G. A. Viano (1969); Coudray and Coz (1970); Jean and Sabatier (1973); I. Miodek (1976); C. Coudray (1978 and 1980); Lipperheide and Fiedeldey (1978 and 1981); Munchow and Scheldt (1980). A more general method was given by P. C. Sabatier (1971 and 1972a). For numerical calculations and applications, see F. Q. van Phu (1970); Sabatier and van Phu (1971); V. V. Malyarov et al. (1975b and 1977); C. Coudray (1977); V. V. Malyarov et al. (1978). The following papers generalized the method to include spin-orbit and tensor forces: P. C. Sabatier (1968); M. A. Hooshyar (1971, 1972, 1975, 1978, 1980); and Coudray and Coz (1972) gave a relativistic generalizations. The Regge approach, working in the complex /-plane, was carried further by G. Burdet et al. (1966); J.-J. Loeffel (1968); Malyarov and Poplavskii (1972); Heniger and Loeffel (1973); V. V. Malyarov et al. (1975c).
The first attempt at solving the full three-dimensional inverse scattering problem for noncentral potentials was made by Kay and Moses (1961a and b) based on their general method of (1955), but with incomplete success. A different approach was taken by L. D. Faddeev (1974) and R. G. Newton (1974a, b) [but see corrections mentioned on p. 334 of Chadan and Sabatier (1977)] on the basis of a new Green’s function introduced by L. D. Faddeev (1965 and 1966). A new and more natural solution was given by R. G. Newton (1979a, 1980b, c, 1981b, 1982). Secs. 20.5.2 and 20.5.3 are based on these papers. See also A. J. Devaney (1978), and H. E. Moses (1980); Y. Saito (1980b and 1981).
The idea of reconstructing the potential from the back-scattering amplitude A(—k,k) goes back to H. E. Moses (1956). His procedure was further developed by R. T. Prosser (1969, 1976, 1980). This Section is based on their work. General. From a practical point of view the question of stability of inversion procedures and the handling of errors in the data, is of great importance. The following papers deal with this problem: V. A. Marchenko (1968); Lundina and Marchenko (1969); Di Salvo and Viano (1976); V. P. Zhigunov (1979); W. W. Symes (1980). Other relevant references are Bernstein and Muckerman (1967); Calogero and Cox (1968); F. J. Dyson (1976); Zakhariev (1976); J. T. Cushing (1977); P. C. Sabatier (1978 and 1982); Gerber and Yinnon (1980).
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Newton, R.G. (1982). The Inverse Scattering Problem. In: Scattering Theory of Waves and Particles. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88128-2_20
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