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We develop {\it ab initio} relativistic QED theory for elastic electronscattering on hydrogen-like highly charged ions for impact energies where, inaddition to direct (Coulomb) scattering, the process can also proceed viaformation and consequent Auger decay of autoionizing states of thecorresponding helium-like ions. Even so the primary goal of the theory is totreat electron scattering on highly charged ions, a comparison with experimentshows that it can also be applied for relatively light ions covering thus avery broad range of the scattering systems. Using the theory we performedcalculations for elastic electron scattering on B$^{4+}$, Ca$^{19+}$,Fe$^{25+}$, Kr$^{35+}$, and Xe$^{53+}$. The theory was also generalized forcollisions of hydrogen-like highly charged ions with atoms considering thelatter as a source of (quasi-) free electrons.

QED theory of elastic electron scattering on hydrogen-like ions involving formation and decay of autoionizing states