  The purpose of this paper is to study the exceptional eigenvalues of the
asymmetric quantum Rabi models (AQRM), specifically, to determine the
degeneracy of their eigenstates. Here, the Hamiltonian
$H^{\epsilon}_{\text{Rabi}}$ of the AQRM is defined by adding the fluctuation
term $\epsilon \sigma_x$, with $\sigma_x$ being the Pauli matrix, to the
Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry.
The spectrum of $H^{\epsilon}_{\text{Rabi}}$ contains a set of exceptional
eigenvalues, considered to be remains of the eigenvalues of the uncoupled
bosonic mode, which are further classified in two types: Juddian, associated
with polynomial eigensolutions, and non-Juddian exceptional. We explicitly
describe the constraint relations for allowing the model to have exceptional
eigenvalues. By studying these relations we obtain the proof of the conjecture
on constraint polynomials previously proposed by the third author. In fact, we
prove that the spectrum of the AQRM possesses degeneracies if and only if the
parameter $\epsilon$ is a half-integer. Moreover, we show that non-Juddian
exceptional eigenvalues do not contribute any degeneracy and we characterize
exceptional eigenvalues by representations of $\mathfrak{sl}_2$. Upon these
results, we draw the whole picture of the spectrum of the AQRM. Furthermore,
generating functions of constraint polynomials from the viewpoint of confluent
Heun equations are also discussed.
