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 ε Rabi of the AQRM is defined by adding the fluctuation term εσ x , with σ x being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its Z 2 -symmetry. The spectrum of H ε 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 ε 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 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.
Abstract.We derive an expression for the value ζ Q (3) of the spectral zeta function ζ Q (s) for the non-commutative harmonic oscillator using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted J 2 (n) and J 3 (n), which can be regarded as analogues of the Apéry numbers, naturally arise and play a key role in obtaining the expressions for the values ζ Q (2) and ζ Q (3). We also show that the numbers J 2 (n) and J 3 (n) have congruence relations such as those of the Apéry numbers.
The Apéry-like numbers J2(n) associated to the special value ζQ(2) of the spectral zeta function ζQ(s) for the non-commutative harmonic oscillator Q have remarkable modular form interpretation. In fact, we show that the differential equation satisfied by the generating function w2(t) of J2(n) is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4-torsion. The parameter t of this family can be regarded as a modular function for the congruent subgroup Γ0(4). Further, we see that the function w2(t) is regarded as a Γ0(4)-modular form of weight 1 in the variable τ by taking t as the classical Legendre modular function λ(τ).
Special values ζQ(k) (k = 2, 3, 4, ...) of the spectral zeta function ζQ(s) of the non-commutative harmonic oscillator Q are discussed. Particular emphasis is put on basic modular properties of the generating function w k (t) of Apéry-like numbers which is appeared in analysis on the first anomaly of each special value. Here the first anomaly is defined to be the "1st order" difference of ζQ(k) from ζ(k), ζ(s) being the Riemann zeta function. In order to describe such modular properties for k ≥ 4, we introduce a notion of residual modular forms for congruence subgroups of SL2(Z) which contains the classical notion of Eichler integrals as a particular case. Further, we define differential Eisenstein series, which are residual modular forms. Using such differential Eisenstein series, for example, one obtains an explicit description of w4(t). A certain Eichler cohomology group associated to such residual modular forms plays also an important role in the discussion.
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