The hidden symmetry of the asymmetric quantum Rabi model (AQRM) with a half-integral bias was uncovered in recent studies by the explicit construction of operators $J_\ell$ commuting with the Hamiltonian. The existence of such symmetry has been widely believed to cause the degeneration of the spectrum, that is, the crossings on the energy curves. In this paper we propose a conjectural relation between the symmetry and degeneracy for the AQRM given explicitly in terms of two polynomials appearing independently in the respective investigations. Concretely, one of the polynomials appears as the quotient of the constraint polynomials that assure the existence of degenerate spectrum while the other determines a quadratic relation (in general, it defines a hyperelliptic curve) between the AQRM Hamiltonian and its basic commuting operator $J_\ell$. The significance of the conjecture is that it provides a concrete and unexpected realization of the presumed relation between the hidden symmetry and the degeneracy of the AQRM with a half-integral bias, and moreover, that the resulting equation leads to structural insights of the whole spectrum. For instance, the energy curves are naturally shown to lie on a surface determined by the family of hyperelliptic curves by considering the coupling constant as a variable. This geometric picture contains the generalization of the parity decomposition of the symmetric quantum Rabi model. Moreover, it allows us to describe a remarkable approximation of the first $\ell$ energy curves by the zero-section of the corresponding hyperelliptic curve. These investigations naturally lead to a geometric picture of the (hyper-) elliptic surfaces given by the Kodaira–Néron type model for a family of energy curves over the projective line, which may be expected to contribute to a complex analytic proof of the conjecture.

中文翻译：

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具有积分偏差的非对称量子Rabi模型的简并和隐对称性

在最近的研究中，通过显式构造与哈密顿量交换的算子 $J_\ell$，揭示了具有半积分偏差的非对称量子拉比模型 (AQRM) 的隐藏对称性。人们普遍认为这种对称性的存在会导致光谱的退化，即能量曲线上的交叉。在本文中，我们提出了 AQRM 的对称性和简并性之间的猜想关系，该关系是根据在各自研究中独立出现的两个多项式明确给出的。具体来说，其中一个多项式显示为确保退化谱存在的约束多项式的商，而另一个确定 AQRM 哈密顿量与其基本交换算子之间的二次关系（通常，它定义超椭圆曲线）$J_\厄尔$。该猜想的意义在于，它提供了一个具体且意想不到的实现，即隐含对称性与具有半积分偏差的 AQRM 的简并性之间的假定关系，此外，由此产生的方程导致了对整个频谱的结构洞察. 例如，通过将耦合常数视为变量，能量曲线自然地显示在由超椭圆曲线族确定的曲面上。这张几何图包含对称量子拉比模型的奇偶分解的推广。此外，它允许我们通过相应的超椭圆曲线的零截面来描述第一个 $\ell$ 能量曲线的显着近似。