We develop {it ab initio} relativistic QED theory for elastic electron scattering on hydrogen-like highly charged ions for impact energies where, in addition to direct (Coulomb) scattering, the process can also proceed via formation and consequent Auger decay of autoionizing states of the corresponding helium-like ions. Even so the primary goal of the theory is to treat electron scattering on highly charged ions, a comparison with experiment shows that it can also be applied for relatively light ions covering thus a very broad range of the scattering systems. Using the theory we performed calculations for elastic electron scattering on B$^{4+}$, Ca$^{19+}$, Fe$^{25+}$, Kr$^{35+}$, and Xe$^{53+}$. The theory was also generalized for collisions of hydrogen-like highly charged ions with atoms considering the latter as a source of (quasi-) free electrons.
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