System of exact equations for the mass operator of quasiparticles interacting with phonons

A diagram technique is used to construct a system of exact equations for the total mass operator of quasiparticles interacting with phonons. Its solution is an infinite integral continued fraction with exactly determined n-th term. For the example of an exactly solvable problem, the spectrum of the ground state and all bound states of a quasiparticle with phonons is found. The part played by many-phonon processes in its formation is investigated for different magnitudes of the coupling constant.