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Spectral transitions in some Rabi models

Grzegorz Świderski, Lech Zieliński

Abstract

We consider the transition from discrete to continuous spectrum which occurs in some quantum Rabi models. The subordinacy theory is used to detect and locate the essential spectrum of the intensity-dependent Rabi model, the anisotropic two-photon Rabi model, and the two-photon Rabi-Stark model for the whole range of parameters. The absence of a singular spectrum and the absence of eigenvalues in the interior of the essential spectrum are proved for all models considered.

Spectral transitions in some Rabi models

Abstract

We consider the transition from discrete to continuous spectrum which occurs in some quantum Rabi models. The subordinacy theory is used to detect and locate the essential spectrum of the intensity-dependent Rabi model, the anisotropic two-photon Rabi model, and the two-photon Rabi-Stark model for the whole range of parameters. The absence of a singular spectrum and the absence of eigenvalues in the interior of the essential spectrum are proved for all models considered.
Paper Structure (34 sections, 12 theorems, 115 equations)

This paper contains 34 sections, 12 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Notations
  4. Intensity-dependent Rabi model
  5. Two-photon Rabi model
  6. Anisotropic two-photon Rabi model
  7. Two-photon Rabi-Stark quantum model
  8. The general scheme
  9. A criterion for self-adjointness
  10. Stolz class
  11. Periodic modulations
  12. Main tools
  13. An auxiliary result
  14. Proofs of Theorems \ref{['thm:21']}-\ref{['thm:24']}
  15. Proof of Theorem \ref{['thm:21']}
  16. ...and 19 more sections

Key Result

Theorem 2.1

The operator $H$ is unitarily similar to the direct sum where $J^\pm =J((a_n ),(b_n^\pm ))$ is the Jacobi operator with The operator $J^\pm$ is self-adjoint in $\ell^2(\mathbb{N}_0)$ and Moreover, in all cases, $\sigma_{\mathrm{sc}}(J^\pm) = \emptyset$, and $\sigma_{\mathrm{p}}(J^\pm)$ is disjoint from the interior of $\sigma_{\mathrm{ac}}(J^\pm)$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1: Carleman
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • ...and 6 more