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Eigenfunction Form of the Nonrelativistic Coulomb Green's Function
1.R. A. Mapleton, J. Math. Phys. 2, 478 (1961). This paper is referred to hereafter as A.
2.N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Clarendon Press, Oxford, England, 1949), 2nd Ed., pp. 53 and 113.
3.The subscripts 1 and 2 on W correspond to (+) and (−) on respectively.
4.Gordon, for instance, gives a similar decomposition which is valid regardless of the sign of Im z. [W. Gordon, Z. Physik, 48, 180 (1928)]. That the same is not the case here may be understood as follows: In the decomposition given by Eq. (7) for Im specifications on the phases are so rigid that no room is left in which to adjust the phases which would reinstate the validity of Eq. (7) for Im
5.Decomposition of this nature is used in R. A. Mapleton, J. Math. Phys. 3, 297 (1962),
5.which refines some of the arguments in his earlier work [R. A. Mapleton, J. Math. Phys. 2, 482 (1961)]. It is to be noted that the W functions defined in these two articles are not exactly the same. The used in the former are what are called here. It is easily seen, however, that the argument in terms of U is analogous to that in terms of W with the replacement of by
6.To be more exact, should appropriately be identified as either or However, for most cases, it is permissible and simpler to use
7.For the relationship between the solutions obtained from and see F. M. Odeh, J. Math. Phys. 2, 794 (1961).
8.See, for example, reference A.
9.The spherical harmonics are those denned in H. A. Bethe and E. E. Salpeter, in Handbuch der Physik, edited by S. Flügge (Springer‐Verlag, Berlin, 1957), Vol. XXXV.
10.See Gordon’s paper in reference 4.
11.A. Sommerfeld, Atombau und Speklrallinien (Frederick Vieweg und Sohn, Braunschweig, Germany, 1939), Vol. 2, p. 457.
12.See, for example, the treatment in the first paper in reference 5.
13.A. Sommerfeld, Ann. Physik 11, 257 (1931), Appendix.
14.See p. 112 of the book cited in reference 2.
15.See reference 11. The functions and are known to be adequate for representing the particle in the initial and final state, respectively. Therefore, it is natural, and in fact simpler, to choose (+) or (−) for (S) in Eq. say, when the diverging or converging solution is desired.
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