  In this work the general results about asymptotics of eigenvalues of
unbounded operators are obtained. We consider here different cases of compact,
relatively compact, selfadjoint or nonselfadjoint perturbations. In particular
we prove a generalization of Janas-Naboko lemma about eigenvalues asymptotics
of unbounded operators at compact perturbation. A generalization of our
previous result about noncompact perturbation of oscillator spectrum is also
given. As an example we consider two-photon quantum Rabi model. We obtain
tree-term asymptotic formula for large eigenvalues of the energy operator of
this model. The asymptotics of related to this model polynomials is found. We
give also an original proof of the Perelomov factorization theorem for
contraction operator of quantum optics.
