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Modification of Effective-Range Theory in the Presence of a Long-Range (r−4) Potential

J. Math. Phys. 2, 491 (1961); doi:10.1063/1.1703735

Issue Date: July 1961

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Thomas F. O'Malley, Larry Spruch, and Leonard Rosenberg
Physics Department, Washington Square College, and Institute of Mathematical Sciences, New York University, New York, New York

Abstract not available.

Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
History: Received March 3, 1961
Permalink: http://link.aip.org/link/?JMAPAQ/2/491/1
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ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (19)
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  1. J. M. Blatt and J. D. Jackson, Phys. Rev. 76, 18 (1949).
  2. See also J. D. Jackson and J. M. Blatt, Revs. Modern Phys. 22, 77 (1950);
    3. G. F. Chew and M. L. Goldberger, Phys. Rev. 75, 1637 (1949).
  3. H. A. Bethe, Phys. Rev. 76, 38 (1949).
  4. With the exception of the spherical Bessel functions jL(kr) and nL(kr), the angular momentum under consideration, when explicitly indicated, will appear in parentheses; subscripts will generally refer to the energy under consideration.
  5. The smallness of the numerical value of some characteristic length, which is after all a dimensional quantity, is not necessarily a measure of the usefulness of effective-range theory. Energies are in many regards “scaled” in relation to the characteristic length, and it is probably true that in general the range of validity of effective-range theory will cover a significant portion of the energy spectrum. It seems fair to say that effective-range theory was introduced in nuclear rather than in atomic theory not because of the disparity in the values of the characteristic lengths, but because of the lack of knowledge of the nuclear potential. To the extent that the nuclear potential can now be considered to be known, and to the extent that we restrict ourselves to scattering by a compound system—these are of course very severe restrictions—effective-range theory, appropriately modified, is potentially as valuable in atomic as in nuclear physics.
  6. A preliminary discussion of the results of the present paper has already appeared. [L. Spruch, T. F. O'Malley, and L. Rosenberg, Phys. Rev. Letters 5, 375 (1960)]. Equation (1) of this reference contains a misprint; in the last term on the right-hand side A2 should be replaced by A3.
  7. Castillejo, Percival, and Seaton, Proc. Roy. Soc. (London) A254, 259 (1960).
  8. E. Gerjuoy and S. Stein, Phys. Rev. 97, 1671 (1955);
    9. B. H. Bransden, A. Dalgarno, T. L. John, and M. J. Seaton, Proc. Phys. Soc. (London) 71, 877 (1958).
  9. R. M. Thaler, Phys. Rev. 114, 827 (1959).
  10. L. M. Delves, Nuclear Phys. (to be published).
  11. E. Vogt and G. H. Wannier, Phys. Rev. 95, 1190 (1954).
    12. Though we have made no use of their results beyond the basic result that one can convert the Schrödinger equation with a 1/r4 term into a Mathieu equation, it may be useful to record the connection between their parameters and ours, namely,

    [dformula beta = nu - L,cos pi gamma = (1 + m[sup 2])/(2m),e[sup phi] = (-1)[sup L](1 - m[sup 2])/(2m sin nu pi)]

    .

  12. J. Meixner and F. W. Schafke, Die Grundlehren der Mathematischen Wissenschaften (Springer-Verlag, Berlin, Germany, 1954), Vol. 71.
  13. The higher terms in the expansion of nu can be obtained from the expansion of cos nupi given by Meixner and Schafke, reference 11, Eq. (2.2-44). These higher terms will not, however, be required for our present purposes.
  14. Equation (5.4) is identical in form with the usual expansion for k2L+1 cot eta(L), vp having been chosen to accomplish this. Equation (4.3) is then rather more complicated in form than it might otherwise be.
  15. The alternate asymptotic expansion of the Mathieu function follows from the realization (see reference 11) that Mnu(1) approaches Jnu, and that there are different asymptotic expansions of Jnu depending upon which sector of the complex plane the argument of Jnu lies in.
  16. L. Spruch and L. Rosenberg, Phys. Rev. 116, 1034 (1959);
    17. 117, 1095 (1960)
    and L. Rosenberg, L. Spruch, and T. F. O'Malley, ibid. 118, 184 (1960).
  17. Thaler's expression (reference 8) for arbitrary L is correct only for L = 1 and for L = 1.
  18. The replacement of k by igamma is permissible because, as noted previously, the Mathieu functions that we have used are valid for −pi<argk<pi; the question of the analyticity of k cot eta(L), for example, does not arise. Equation (6.4), incidentally, is formally equivalent to replacing tan eta(L) by i in Eq. (4.3).
  19. T. Ohmura, Y. Hara, and T. Yamanouchi, Progr. Theoret. Phys. (Kyoto) 22, 152 (1959);
    20. 20, 80 (1958);
    T. Ohmura and H. Ohmura, Phys. Rev. 118, 154 (1960).
  20. L. Rosenberg, L. Spruch, and T. F. O'Malley, Phys. Rev. 119, 164 (1960).

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