QED calculations of three-photon transition probabilities in H-like ions with arbitrary nuclear charge

The quantum electrodynamical theory of the three-photon transitions in hydrogen-like ions is presented. Emission probabilities of various three-photon decay channels for $2p_{3\mathrm{/}2}$, $2p_{1\mathrm{/}2}$ and $2s_{1\mathrm{/}2}$ states are calculated for the nuclear charge $Z$ values $1⩽Z⩽95$. The results are given in two different gauges. The fully relativistic three-photon decay rates of hydrogen-like ions with half-integer nuclear spin are given for transitions between fine structure components. The results can be applied to the tests of the Bose-Einstein statistics for the multiphoton systems.