Opuscula Math. 41, no. 3 (2021), 301-333

Opuscula Mathematica

Perturbation series for Jacobi matrices and the quantum Rabi model

Mirna Charif
Lech Zielinski

Abstract. We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.

Keywords: Jacobi matrix, unbounded self-adjoint operators, quasi-degenerate eigenvalue perturbation, perturbation series, quantum Rabi model, rotating wave approximation.

Mathematics Subject Classification: 81Q10, 47B36, 15A18.

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  • Mirna Charif
  • ORCID iD https://orcid.org/0000-0002-4403-1453
  • Université du Littoral Côte d’Opale, Laboratoire de Mathématiques Pures et Appliquées, Joseph Liouville EA 2597, F-62228 Calais, France
  • Lebanese University, Faculty of Sciences, Department of Mathematics, P.O. Box 826 Tripoli, Lebanon
  • Lech Zielinski (corresponding author)
  • ORCID iD https://orcid.org/0000-0002-7314-7586
  • Université du Littoral Côte d’Opale, Laboratoire de Mathématiques Pures et Appliquées, Joseph Liouville EA 2597, F-62228 Calais, France
  • Communicated by P.A. Cojuhari.
  • Received: 2020-12-02.
  • Revised: 2021-01-17.
  • Accepted: 2021-01-23.
  • Published online: 2021-04-19.
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Cite this article as:
Mirna Charif, Lech Zielinski, Perturbation series for Jacobi matrices and the quantum Rabi model, Opuscula Math. 41, no. 3 (2021), 301-333, https://doi.org/10.7494/OpMath.2021.41.3.301

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