Off-diagonal properties of the Feynman propagator and the Green function for a bare Coulomb field

Blinder [Phys. Rev. A, 43, 13 (1991)] has given an explicit diagonal form for the nonrelativistic Coulomb propagator. His result is an infinite series involving Whittaker, Laguerre, Hermite, and error functions. However, he notes that at the nucleus, only error functions remain in the infinite series. Here, an off-diagonal generalization of Blinder’s result at the nucleus is constructed. By relating this result to the Green function, it is then shown that one is led in this latter representation to an explicit sum of Blinder’s infinite series. Finally, contact is established between the present off-diagonal study and the recent work of Chibisov et al. [Phys. Rev. Lett. 84, 451 (2000)] on the Coulomb Green function.

Figure 1

The $O\left(Z\right)$ term [Eq. (6)] $K_1(r,0,\beta )$ in the March-Murray expansion for the propagator $K(r,0,\beta )$ for the Coulomb potential. Results are shown here for $Z=1$ and for the values $\beta =0.01$ and 0.025

Figure 2

Equation (14) for the Laplace transform of the remainder term $R(0,0,\beta ).$ The case $Z=1$ is shown.

Figure 3

(a) The quantity ${\left(2\pi \beta \right)}^{3∕2}R(0,0,\beta )$ for $Z=1,$ evaluated using Widder’s formula for the inverse Laplace tranform of Eq. (14), with $n=200;$ (b) ${\left(2\pi \beta \right)}^{3∕2}K(0,0,\beta )$ for $Z=1$, evaluated from Eq. (19). The small-$\beta$ and large-$\beta$ approximate forms of Eqs. (20, 23) at $r=0$ are also shown for comparison.