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<p>The Quantum Rabi Hamiltonian (QRM) is a simple model of the interaction between a two-level atom and a single quantized mode of light. Juddian eigenvalues of the QRM are those eigenvalues for which both the eigenvalue and the corresponding eigenvector admit a simple form. We prove a strong form of the density conjecture of Reyes-Bustos and Wakayama, showing that for any fixed value of the splitting between the two atomic levels, there is a dense set of coupling strengths for which the corresponding QRM admits Juddian eigenvalues. We also construct infinitely many sets of parameters for which the QRM admits two distinct Juddian eigenvalues. The fine structure of the zeros of classical Laguerre polynomials plays a key role in our methods.</p>

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The density conjecture for Juddian points for the quantum Rabi model

  • 期刊论文

  • 发表日期：

    2025年12月3日

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  • Rishi Kumar,
  • Zeév Rudnick

摘要

The Quantum Rabi Hamiltonian (QRM) is a simple model of the interaction between a two-level atom and a single quantized mode of light. Juddian eigenvalues of the QRM are those eigenvalues for which both the eigenvalue and the corresponding eigenvector admit a simple form. We prove a strong form of the density conjecture of Reyes-Bustos and Wakayama, showing that for any fixed value of the splitting between the two atomic levels, there is a dense set of coupling strengths for which the corresponding QRM admits Juddian eigenvalues. We also construct infinitely many sets of parameters for which the QRM admits two distinct Juddian eigenvalues. The fine structure of the zeros of classical Laguerre polynomials plays a key role in our methods.