Abstract
We now assume that the interaction is local and spherically symmetric, i.e., described by a potential function of r = |r| only,
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Notes and References
The partial-wave expansion of the scattering amplitude was introduced into quantum mechanics by Faxen and Holtsmark (1927). It is, of course, patterned after its previous use in the theory of sound and electromagnetic waves. A. D. Boardman et al. (1967) present a partial wave expansion for noncentral potentials. The use of Padéapproximants to sum the partial wave series is dealt with by J. Fleischer (1972 and 1973); K. Alder et al. (1973); O. D. Corbella et al. (1976); Common and Stacey (1978); F. F. Grinstein (1980). For other partial wave expansions, see E. G. Kalnins et al. (1973); Daumens and Perroud (1979); and for relativistic generalizations, J. Bystřičky et al. (1976); M. Daumens et al. (1979); Daumens and Winternitz (1980). The two-dimensional analog can be found in I. R. Lapidus (1982).
The following papers deal with the issue of time delay in quantum-mechanical two-particle scattering: B. A. Lippmann (1966); Jauch and Marchand (1967); J. L. Agudin (1968); H. M. Nussenzveig (1972); J. M. Jauch et al. (1972); Bollé and Osborn (1975); Ph.A. Martin (1975); Nowakowski and Osborn (1975); Tsang and Osborn (1975); Ph.A. Martin (1976); R. Lavině (1978); D. Bollé (1979a); J. O. Hirschfelder (1979); H. Narnhofer (1980); see also MacMillan and Osborn (1980). For time delay in n-particle scattering, see Osborn and Bollé (1974, 1975, 1976, 1978); Bollé and Osborn (1979); S. Bosanac (1981a); and in classical particle scattering, T. A. Osborn et al. (1980); Narnhofer and Thirring (1981).
The effective-range theory is due to J. Schwinger (1947) and his unpublished lecture notes [see Blatt and Jackson (1949)]. The method given here is due to H. Bethe (1949). See also Jackson and Blatt (1950). The fact that the introduction of a new bound state causes the scattering length to be infinite has been exploited for the numerical calculation of those potential strengths at which new bound states of various angular momenta appear by Schey and Schwartz (1965) for Yukawa potentials as well as for Wood-Saxon potentials. For investigations of the low-energy behavior of the scattering amplitude in the case of a long-range potential, when the effective-range theory breaks down, see Spruch, O’Malley, and Rosenberg (1960 and 1962); Levy and Keller (1963); Hinkelman and Spruch (1971). The most detailed study of the Schrödinger equation for the square-well potential is that of H. M. Nussenzveig (1959); see also S. Tani (1968). Although the Levinson theorem was known to a number of persons before, it was first proved rigorously by N. Levinson (1949b). The Breit-Wigner resonance formula is the quantum-mechanical analog of the electromagnetic Lorentz resonance formula. The two line shapes are identical. The original reference in the present context is Breit and Wigner (1936). The lower limit on the phase-shift derivative, due to causality arguments, as expressed in (11.57), was first given by E. P. Wigner (1955). A relativistically invariant partial-wave analysis was discussed by H. Munczek (1963).
The variational formulation of scattering theory was independently introduced by Hulthén and Schwinger: L. Hulthén (1944, 1947, and 1948); J. Schwinger (1947a, b, and 1950); Lippmann and Schwinger (1950). The general method based on the integral equation is Schwinger’s, and so are specifically Eqs. (11.68) and (11.70). Eq. (11.69) is due to B. A. Lippmann (1956). Hulthén’s procedure uses differential equations, as in Sec. 11.3.3 See also I. Tamm (1948 and 1949); J. M. Blatt (1948) ; W. Kohn (1948 and 1951); Blatt and Jackson (1949); S. Huang (1949); M. Verde (1949); Hulthén and Olsson (1950); T. Kato (1950b); Massey and Moiseiwitch (1951); J. L. Jackson (1951); Troesch and Verde (1951); H. E. Moses (1953b and 1957); Borowitz and Friedman (1953); Turner and Makinson (1953); Boyet and Borowitz (1954); Moe and Saxon (1958); C. Joachain (1965a and b); C. Schwartz (1966); Y. Hahn (1968b); H. Morawitz (1970); Sloan and Brady (1972); Rabitz and Conn (1973); Carew and Rosenberg (1973); Oberoi and Nesbeth (1973); L. Rosenberg (1973); D. G. Truhlar et al. (1974); Rosenberg and Spruch (1974); Singh and Stauffer (1975); Madan and Blankenbecler (1978); R. K. Nesbet (1978 and 1980); Darewych and Pooran (1978); M. Kawai et al. (1978); Spruch and Rosenberg (1978); Darewych and Sokoloff (1979); K. Takatsuka et al. (1981); Takatsuka and McKoy (1981); R. Goldflam et al. (1981). For similar methods in problems of neutron diffusion, see R. E. Marshak (1947); B. Davison (1947); in electromagnetic problems, Levine and Schwinger (1948 and 1949); Kalikstein and Spruch (1964). The development of bounds, or extremal principles, for scattering parameters was started by T. Kato (1951b) and considerably expanded in the following papers: T. Kikuta (1953b); V. Risberg (1956); I. C. Percival (1957 and 1960); Spruch and Kelly (1958); L. Spruch (1958 and 1962); Spruch and Rosenberg (1959 and 1960a and b); Rosenberg and Spruch (1960, 1961, and 1962); Rosenberg, Spruch, and O’Malley (1960a); Bartram and Spruch (1962); Hahn, O’Malley, and Spruch (1962, 1963, and 1964); Y. Hahn (1965); L. Rosenberg (1965 and 1970); Hahn and Spruch (1967).
The stationary formula (11.80) is due to W. Kohn (1948). See also F. E. Harris (1967); R. K. Nesbet (1968); Brownstein and McKinley (1968); Y. Tikochinsky (1970); Takatsuka and Kueno (1979). For variational bounds, see I. Aronson et al. (1967), J. N. Bardsley et al. (1972); K. J. Miller (1971); R. Blau et al. (1975); Darewych and Pooran (1980).
The arguments leading to the bounds (11.87), (11.90), and (11.93) follow Y. Hahn et al. (1963). See also A. M. Arthurs (1968 and 1975); D. Gelman (1974); Hahn and Spruch (1974); R. Blau et al. (1974 and 1977); Rosenberg and Spruch (1975); I. Aronson et al. (1979). The Hylleraas-Undheim theorem is due to Hylleraas and Undheim (1930). The theorem is a variant of facts that are well known in algebra and referred to in this paper. The proof given here follows J. K. L. MacDonald (1933). The following papers apply variational methods to resonances: A. Ronveau (1968b); Y. Hahn (1972).
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Newton, R.G. (1982). Single-Channel Scattering of Spin 0 Particles, I. 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_11
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