Theory of the multiphoton cascade transitions with two photon links: comparison of Quantum Electrodynamical and Quantum Mechanical approaches

QED theory of multiphoton cascade transitions in atoms and ions is developed. In particular the $3s\to 1s+2\gamma$, $4s\to 1s+2\gamma$ and $3p\to 1s+3\gamma$ processes are considered. Two different approaches (Quantum Electrodynamical and phenomenological Quantum Mechanical) are applied.