Український математичний конгрес  2009
Olga Khetselius (Odessa State Environmental University, Odessa, Ukraine) Green’s function method in quantum theory: New numerical algorithm for the Dirac equation with complex energy and Fermimodel nuclear potential We present a new effective approach to construction of the electron Green function for the Dirac equation with a nonsingular central nuclear Fermimodel 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 multielectron system accounts for multibody correlation effects. A minimization of the functional ImdE leads to integraldifferential KohnShamlike 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 Fermimodel nuclear potential and equations for calculating the Slater type integrals in the Mohr formula for definition of the selfenergy 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 selfenergy radiative correction to the atomic levels energies [3].
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