Matrix elements with respect to a discrete ‘‘Sturmian’’ basis set of the operator r1/2gr1/2, where g is the Coulomb Green’s function of the ‘‘second‐order’’ Dirac equation, are investigated. Closed expressions, involving Gaussian hypergeometric functions, can be obtained for the matrix elements. This calculation illustrates contour integration techniques capable of yielding closed expressions for a number of overlap integrals associated with the relativistic Kepler problem.
REFERENCES
1.
2.
3.
4.
5.
N. L.
Manakov
, L. P.
Rapoport
, and S. A.
Zapryagaev
, Phys. Lett. A
43
, 139
(1973
);N. L.
Manakov
, L. P.
Rapoport
, and S. A.
Zapryagaev
, J. Phys. B
7
, 1076
(1974
);A. I.
Mil’shtein
and V. M.
Strakhovenko
, Phys. Lett. A
90
, 447
(1982
).6.
7.
8.
9.
10.
11.
12.
L. M. Brown, “Two‐component fermion theory,” in Lectures in Theoretical Physics (Interscience, New York, 1962), Vol. IV.
13.
L. M. Brown, “Quantum electrodynamics at high energy,” in Topics in Theoretical Physics, Proceedings of the Liperi Summer School in Theoretical Physics 1967, edited by C. Cronstrom (Gordon and Breach, New York, 1969), p. 113.
14.
15.
16.
17.
18.
19.
20.
21.
H. Buchholz, Die Konfluente Hypergeometrische Function (Springer, Berlin, 1953), p. 135, Eq. (2).
22.
F. Klein, Vorlesungen ueber die Hypergeometrische Funktion (Springer, Berlin, 1933).
23.
See Ref. 4, Eq. (2.21).
24.
An integral representation for a product of two Whittaker functions with different arguments is needed [Ref. 21, Eq. (5c), p. 86; the condition Re(μ)>0stated in Buchholz is a misprint, and should read Re(r)>0];as well as an integral representation for the Bessel function, [G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1962), 2nd ed., p. 177, Eq. (8), and the relation on p. 77:
25.
See Ref. 21, p. 137, Eq. (10b).
26.
See Ref. 21, p. 136, Eq. (9).
27.
See Ref. 21, p. 137, Eq. (10g) implies the relationship or This last identity used two times leads to Eq. (A7) in the Appendix.
This content is only available via PDF.
© 1987 American Institute of Physics.
1987
American Institute of Physics
You do not currently have access to this content.
