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Theory of the spectrum of localized two-level quasi-particle renormalized dueto interaction with polarization phonons at cryogenic temperature is developedusing the generalized method of Feynman-Pines diagram technique. Using theprocedure of partial summing of infinite ranges of the main diagrams, massoperator is obtained as a compact branched chain fraction, which effectivelytakes into account multi-phonon processes. It is shown that multi-phononprocesses and interlevel interaction of quasiparticle and phonons cardinallychange the renormalized spectrum of the system depending on the difference ofenergies of two states, which either resonates with phonon energy or does not.The spectrum of non-resonant system contains renormalized energies of the mainstates and two similar infinite series of groups of phonon satellite levels.The spectrum of a resonant system contains a renormalized ground state andinfinite series of satellite groups.

condensed matter physics

Generalized method of Feynman-Pines diagram technique in the theory of  energy spectrum of two-level quasiparticle renormalized due to multi-phonon  processes at cryogenic temperature