Fourier and microlocal methods for quantum Rabi systems

We give a concise presentation two perturbative approaches to the spectral analysis of Quantum Rabi systems. For the one-mode two-level model, a metaplectic translation reduces the Hamiltonian to a bounded analytic perturbation of a doubled harmonic oscillator. The first-order effective matrix is computed by Fourier-Hermite methods and its entries are expressed through Laguerre polynomials. This gives analytic spectral branches, parity information, and the validity of Braak's conjecture on arbitrary fixed finite spectral windows for a sufficiently small atomic gap. We also discuss multi-level, multi-mode Rabi systems, for $n=2$ and $N=3$ . Although their semiprincipal symbols need not be smoothly diagonalizable, an order-one regularizing perturbation places them in a globally diagonalizable class; comparison by the minimax principle then yields a two-term Weyl asymptotic formula. Complete proofs and additional computations appear in [12]; here the emphasis is on the Fourier and microlocal techniques behind the results.