论文标题
覆盖不对称量子狂犬模型的模型：$η$降低非交通谐波振荡器
Covering models of the asymmetric quantum Rabi model: $η$-shifted non-commutative harmonic oscillators
论文作者
论文摘要
非交通谐波振荡器（NCHO）是一个矩阵的差差算子，最初是作为具有较弱$ \ mathfrak {sl} _2（\ Mathbb {r}）量的量子谐波振荡器的概括性引入的。 NCHO的光谱具有显着的特性，包括存在数字理论结构的存在，例如模块化形式，椭圆形曲线和Eichler共同体，在相关光谱Zeta功能的特殊值中观察到。此外，NCHO的特征值问题的Heun Ode图片揭示了与量子光学量基的基本相互作用模型的量子兔模型（QRM）的联系。在本文中，我们引入了$η$降低的NCHO（$η$ -NCHO），该NCHO（$η$ -NCHO）与非对称量子Rabi模型（AQRM）有类似的关系并描述其基本属性。即使移位因子并没有打破NCHO的均等对称性，但在\ frac12 \ mathbb {z} $中出现了某种类型的脱落性，就好像在反映了AQRM的情况一样。此外，我们对汇合过程进行了详细的描述，我们称之为ISO-Parallel Confluence过程，因为它需要并行转换两个参数，描述了$η$ -NCHO的频谱以及$ \ Mathfrak {sl} _2 _2 _2 _2 _2（\ Mathbb {r}）$。我们将两个模型的特征值在ISO-Paralallel汇合过程下，包括$η$ -NCHO的准表演特征函数如何对应于AQRM的Juddian解决方案。从这个汇合过程的角度来看，$η$ -ncho的家族对应于单个AQRM，因此我们可以将$η$ -ncho视为AQRM的覆盖率。我们希望从这种角度开始对$η$ -NCHO和AQRM进行研究，这有助于阐明有关AQRM上的几个问题，包括隐藏的对称性和犹太人解决方案的数量。
The non-commutative harmonic oscillator (NCHO) is a matrix valued differential operator originally introduced as a generalization of the quantum harmonic oscillator having a weaker $\mathfrak{sl}_2(\mathbb{R})$-symmetry. The spectrum of the NCHO has remarkable properties, including the presence of number theoretical structures such as modular forms, elliptic curves and Eichler cohomology observed in the special values of the associated spectral zeta function. In addition, the Heun ODE picture of the eigenvalue problem of the NCHO reveals a connection with the quantum Rabi model (QRM), a fundamental interaction model from quantum optics. In this paper we introduce an $η$-shifted NCHO ($η$-NCHO) that has an analogous relation with the asymmetric quantum Rabi model (AQRM) and describe its basic properties. Even though the shift factor does not break the parity symmetry of the NCHO, a certain type of degeneracies appears for $η\in \frac12 \mathbb{Z}$, as if mirroring the situation of the AQRM. We give furthermore a detailed description of the confluence process, that we call iso-parallel confluence process due to the fact that it requires a parallel transformation of two parameters describing the spectrum of $η$-NCHO and representations of $\mathfrak{sl}_2(\mathbb{R})$. We relate the eigenvalues of the two models under the iso-parallel confluence process, including how the quasi-exact eigenfunctions of the $η$-NCHO correspond to Juddian solutions of the AQRM. From the point of view of this confluence process, a family of $η$-NCHO corresponds to a single AQRM, thus we may regard the $η$-NCHO as a covering of the AQRM. We expect the study of the $η$-NCHO and the AQRM from this point of view to be helpful for the clarification of several questions on the AQRM, including the hidden symmetry and the number of Juddian solutions.
