Perturbation Theory and the Quantum Rabi-model

In the first part of the paper we study a perturbative model of the Rabi system of Quantum Optics. We are therefore able to describe, through Rellich's theory, an analytic expansion of finite families of eigenvalues, of arbitrary fixed length. In particular, we prove that for finite families of eigenvalues the Braak conjecture holds. In the second part we study the asymptotics of the Weyl spectral counting function of a class of systems that generalize the Quantum Rabi Model to an $N$-level atom ($N\geq3$) with $n=N-1$ cavity modes of the electromagnetic field, with particular interest in the case $N=3$ and $n=2$.