{
  "id": "2008.05354",
  "version": "v1",
  "published": "2020-08-12T14:51:48.000Z",
  "updated": "2020-08-12T14:51:48.000Z",
  "title": "Heat kernel for the quantum Rabi model II: propagators and spectral determinants",
  "authors": [
    "Cid Reyes-Bustos",
    "Masato Wakayama"
  ],
  "comment": "22 pages, 2 figures. The section on the meromorphic continuation of spectral zeta function was originally part of arXiv:1906.09597, but it was separated and integrated into this article",
  "categories": [
    "math-ph",
    "math.MP",
    "math.NT",
    "quant-ph"
  ],
  "abstract": "The quantum Rabi model (QRM) is widely recognized as an important model in quantum systems, particularly in quantum optics. The Hamiltonian $H_{\\text{Rabi}}$ is known to have a parity decomposition $H_{\\text{Rabi}} = H_{+} \\oplus H_{-}$. In this paper, we give the explicit formulas for the propagator of the Schr\\\"odinger equation (integral kernel of the time evolution operator) for the Hamiltonian $H_{\\text{Rabi}}$ and $H_{\\pm}$ by the Wick rotation (meromorphic continuation) of the corresponding heat kernels. In addition, as in the case of the full Hamiltonian of the QRM, we show that for the Hamiltonians $H_{\\pm}$, the spectral determinant is, up to a non-vanishing entire function, equal to the Braak $G$-function (for each parity) used to prove the integrability of the QRM. To do this, we show the meromorphic continuation of the spectral zeta function of the Hamiltonians $H_{\\pm}$ and give some of its basic properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-12T14:51:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "11M41",
      "47D06"
    ],
    "keywords": [
      "quantum rabi model",
      "heat kernel",
      "spectral determinant",
      "hamiltonian",
      "propagator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}