Opposite-parity degeneracy constraints

Derive and analyze the constraint conditions governing opposite-parity spectral degeneracies of the two-photon asymmetric quantum Rabi model, including a complete characterization of the corresponding degenerate solutions.

Background

The paper characterizes shared-parity degeneracies in detail but treats opposite-parity degeneracies primarily through their known spectral form and confluent-hypergeometric representations. A full analysis would require studying the constraint conditions for these degeneracies.

References

We note that a full study of the opposite-parity spectral degeneration requires a detailed study of the constraint conditions for the degenerate solutions described in , we leave its details for a future study.

— Two-photon quantum Rabi models - Spectral degeneracy and symmetries  (2609.00750 - Reyes-Bustos et al., 1 Sep 2026) in End of Section 7.2, “Opposite-parity degeneracies”

For instance, for the elliptic curve case it is an open direction to study the associated L-functions and integral points, and their possible relation with the spectrum of the 2pAQRM and in particular with the special values of its spectral zeta function, i.e., Dirichlet series.

— Two-photon quantum Rabi models - Spectral degeneracy and symmetries  (2609.00750 - Reyes-Bustos et al., 1 Sep 2026) in Section 5.2, paragraph following Problem 5.2