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The purpose of this paper is to study the exceptional eigenvalues of theasymmetric quantum Rabi models (AQRM), specifically, to determine thedegeneracy of their eigenstates. Here, the Hamiltonian$H^{\epsilon}_{\text{Rabi}}$ of the AQRM is defined by adding the fluctuationterm $\epsilon \sigma_x$, with $\sigma_x$ being the Pauli matrix, to theHamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry.The spectrum of $H^{\epsilon}_{\text{Rabi}}$ contains a set of exceptionaleigenvalues, considered to be remains of the eigenvalues of the uncoupledbosonic mode, which are further classified in two types: Juddian, associatedwith polynomial eigensolutions, and non-Juddian exceptional. We explicitlydescribe the constraint relations for allowing the model to have exceptionaleigenvalues. By studying these relations we obtain the proof of the conjectureon constraint polynomials previously proposed by the third author. In fact, weprove that the spectrum of the AQRM possesses degeneracies if and only if theparameter $\epsilon$ is a half-integer. Moreover, we show that non-Juddianexceptional eigenvalues do not contribute any degeneracy and we characterizeexceptional eigenvalues by representations of $\mathfrak{sl}_2$. Upon theseresults, we draw the whole picture of the spectrum of the AQRM. Furthermore,generating functions of constraint polynomials from the viewpoint of confluentHeun equations are also discussed.

Determinant expressions of constraint polynomials and the spectrum of the asymmetric quantum Rabi model