Scattering Theory of Waves and Particles pp 228-259 | Cite as

# Formal Methods of Solution and Approximations

Chapter

## Abstract

We shall now discuss formally several standard methods of solving the scattering problem. These amount to procedures for constructing the complete Green’s function, or the resolvent. Although they are in principle exact, they give rise in a natural manner to approximation methods.

## Keywords

Unit Circle Formal Method Born Approximation Fredholm Determinant Positive Real Axis
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## Notes and References

- 9.1.1The original reference for the
*Born series*and the*Born approximation*is M. Born (1926). Discussions of its convergence, or of the analyticity of the resolvent, are T. Kato (1949 and 1950a); W. Kohn (1952 and 1954); T. Kikuta (1953a); Zemach and Klein (1958); Aaron and Klein (1960); H. Davies (1960); W. Hunziker (1961); K. Meetz (1962); M. Rotenberg (1963); S. Weinberg (1963b and 1964a); F. Coester (1964a); C. Lovelace (1964b); Scadron, Weinberg, and Wright (1964); S. Tani (1965); J. V. Corbett (1968); W. G. Faris(1971); H. Rabitz(1972); P. J. Bushell(1972); Graves-Morris and Rennison (1974).Google Scholar - The trick of using the kernel defined in (9.6) instead of
*K*was introduced by J. Schwinger (1961) and for the present purpose by M. Scadron et al. (1965). It is intimately related to the methods of K. Meetz (1962), who uses the theory of polar kernels, and of F. Coester (1964). For the original mathematical analysis of*polar*kernels, see D. Hilbert (1912); E. Garbe (1915).Google Scholar - The discussion of the behavior of the eigenvalues
*α*(*E*) of*K*(*E*) as functions of*E*is based on the work of S. Weinberg (1963 and 1964a). The present discussion of the analyticity of α(*E*) goes somewhat beyond his. The analytic continuation of the completeness relation is an extension of that of Meetz.Google Scholar - For a variational formulation of the eigenvalues
*α*(*E*) see Wright and Scadron (1964), and Y. Hahn (1965), appendix C.Google Scholar - The following papers deal with rearrangements of the Born series for the purpose of more rapid convergence: M. Rotenberg (1963); M. Wellner (1963); S. Weinberg (1964b).Google Scholar
- The authority sometimes adduced for Eq. (9.17) is a theorem by A. Herglotz (1911). Functions that can be written in that form are called
*Herglotz functions*. They are intimately related to Wigner’s*R*functions; see E. P. Wigner (1951 and 1952a and b); Wigner and Von Neumann (1954).Google Scholar - 9.1.2The paper by R. H. Dalitz (1951) discusses higher Born approximations. For the second Born approximation to relativistic Coulomb scattering, see McKinley and Feshbach (1948). For an examination of the Born approximation at low energies, see P. Swan (1963a). See also Calogero and Charap (1964); A. C. Yates (1979).Google Scholar
- 9.1.3For discussions and applications of the distorted-wave Born approximation see, for example, H. S. W. Massey (1956a); Bassel and Gerjuoy (1960); L. Rosenberg (1964a). For other distorted-wave approximations, see P. Swan (1960,1961, and 1963b); Austern and Blair (1965) and references therein.Google Scholar
- 9.1.4The idea of this section is due to J. Mazo (1964), unpublished.Google Scholar
- 9.2See, for example, Courant and Hilbert (1953), p. 155. The treatment here is based on S. Weinberg (1963a and b and 1964a and b), who resurrected the Schmidt method and, for the physical reason given, called it the
*quasi-particle method*. See also Scadron and Weinberg (1964); T. Sasakawa (1963); Hahn and Luddy (1981).Google Scholar - 9.3For exhibitions of the Fredholm method, see any book on integral equations, for example, Whittaker and Watson (1948), chap. 11; Courant and Hilbert (1953), chap. 3; Morse and Feshbach (1953), chaps. 8 and 9; F. Smithies (1958).Google Scholar
- The Fredholm method was first applied to scattering problems by Jost and Pais (1951).Google Scholar
- The recursion relations for the terms in the Fredholm expansion in terms of traces were first given by J. Plemelj (1904). The present treatment follows R. G. Newton (1961a).Google Scholar
- Hadamard’s inequality can be found in most of the books already mentioned. It is needed in the
*complex*form, as stated in F. Smithies (1958).Google Scholar - The modification of the Fredholm formulas consisting in replacing the elements on the main diagonal by zeros was introduced by O. Kellogg (1902). Equations (9.84) and (9.85) are there credited to a mathematician named Haskins. The entire procedure was referred to by Hubert as “pulling the poison tooth” of the Fredholm determinant (private communication from R. Jost). Other relevant mathematical papers on Fred-holm equations and their solutions are J. Tamarkin (1926); Hille and Tamarkin (1930, 1931, 1934); C. van Winter (1971).Google Scholar
- For an application of Fredholm theory to the iterated Lippmann-Schwinger equation, see N. N. Khuri (1957). See also J. Schwinger (1954); M. Baker (1958); I. Manning (1965); W. P. Reinhardt (1970).Google Scholar
- 9.4The content of this section is based on R. G. Newton (1963b). For other perturbation treatments see Gammel and McDonald (1966) (on Padé approximants); R. Blankenbecler (1966); B. Michalik (1968); I. Manning (1968); Saskawa and Sawada (1977a).Google Scholar

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