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\rightarrow1s+2\gamma $, $ 4s\rightarrow1s+2\gamma $ and $ 3p\rightarrow1s+3\gamma $ processes are considered. Two different approaches (Quantum Electrodynamical and phenomenological Quantum Mechanical) are applied.