**5. Conclusions**

In the present work, it was shown that for different relations between interaction parameters in the final state, appearing after photoionization, different approaches of MBPT can be used. A spectral function approach with account for the imaginary part of the self-energy in simplified fourth-order make it possible to calculate asymmetrical lineshapes of satellites in photoelectron spectra and of low-energy Auger spectra. Dyson's equation in the third-order makes it possible to calculate the line positions in the case of weak interaction between excited states. In particular, in the case of 3s-spectra of 3d-elements accounting for a very limited number of Feynman diagrams made it possible to reach agreemen<sup>t</sup> with the experiment and to draw a qualitative conclusion on many-electron effects for 3d-shell more than half-filled. However, in the case of strong CI in 5p-spectra, the method of secular matrix solution is more appropriate. This series of research was inspired by Miron Ya. Amusia, who was the first person to proposed to use unified MBPT technique for CI and satellite excitations.

**Author Contributions:** V.G.Y., software and theoretical calculations; Y.A.T., methodology and experimental investigations. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work was supported by IGIC RAS state assignment and RRC "KI" state assignment.

**Institutional Review Board Statement:** The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board of IGIC RAS in agreemen<sup>t</sup> with RRC "KI" 04.05.2022.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
