Green’s function method in quantum theory: New numerical algorithm for the Dirac equation with complex energy and Fermi-model nuclear potential

We present a new effective approach to construction of the electron Green function for the Dirac equation with a non-singular central nuclear Fermi-model potential and complex energy [1]. We represent the radial Green function as a combination of two fundamental solutions of the Dirac equation. The approach proposed includes a procedure of generating the relativistic electron functions  with performance of the gauge invariance principle. In order to reach the gauge invariance principle performance we use earlier developed QED perturbation theory approach. In the fourth order of the QED perturbation theory (PT) there are diagrams, whose contribution into imaginary part of radiation width ImdE for the multi-electron system accounts for multi-body correlation effects. A minimization of the functional ImdE leads to integral-differential Kohn-Sham-like density functional equations. Further check for the gauge principle performance is realized by means of the Ward identities [2]. In the numerical procedure we use the effective algorithm, within which a definition of the Dirac equation fundamental solutions is reduced to solving the single system of the differential equations. This system includes also the differential equations for the Fermi-model nuclear potential and equations for calculating the Slater type integrals in the Mohr formula for definition of the self-energy shift to atomic levels energies. Such a approach allows to compensate a main source of the errors, connected with numerical integration on energy and summation on χ in the Mohr expressions during calculating the self-energy radiative correction to the atomic levels energies [3].

References:
1. A.V. Glushkov, Relativistic Quantum Theory, Odessa, Astroprint (2008).-900P.
2. A.V.Glushkov, Energy Approach to Resonance states of compound super-heavy nucleus and EPPP in heavy nucleus collisions//In: Low Energy Antiproton Phys. (AIP) 796, 206 (2005);
3. A. Glushkov, O. Khetselius, L. Lovett, A. Loboda, D. Sukharev, The Green's function of the Dirac equation with complex energy and Fermi-model nuclear potential: New numerical algorithm// Int. J. Quant. Ch., 109 (2009) 1617-1624.