  Theory of the spectrum of localized two-level quasi-particle renormalized due
to interaction with polarization phonons at cryogenic temperature is developed
using the generalized method of Feynman-Pines diagram technique. Using the
procedure of partial summing of infinite ranges of the main diagrams, mass
operator is obtained as a compact branched chain fraction, which effectively
takes into account multi-phonon processes. It is shown that multi-phonon
processes and interlevel interaction of quasiparticle and phonons cardinally
change the renormalized spectrum of the system depending on the difference of
energies of two states, which either resonates with phonon energy or does not.
The spectrum of non-resonant system contains renormalized energies of the main
states and two similar infinite series of groups of phonon satellite levels.
The spectrum of a resonant system contains a renormalized ground state and
infinite series of satellite groups.
