Eigenvalue Spectra of Rabi Models with Infinite Matrix Representations

We investigate the relationship between confluent Heun functions and the eigenvalue spectra of infinite matrices related to the semi-classical and quantum Rabi models, revealing distinct connections in each case. In the semi-classical model, the eigenvalues are explicitly expressed through confluent Heun functions, whereas in the quantum Rabi model, they are determined by zeros of a condition involving confluent Heun functions. Our findings establish a unified framework for solving the eigenvalue problem of infinite-dimensional unbounded matrices related to the Rabi models. We derive some new identities for confluent Heun functions, enabling simplifications and broader applications in mathematics and physics. The explicit eigenvalue expressions in the semi-classical case align with approximate results from earlier studies, while the derived conditions for the quantum model provide a concise and unified form, encompassing special cases that are typically treated as exceptions. We also discuss the energy spectrum of the quantum Rabi model, uncovering intriguing phenomena and patterns. Our results deepen the understanding of Rabi models and extend their potential applications in quantum optics and quantum information.