标题： 精确求和规则和从微分方程/积分模型对应关系得出的Zeta生成公式 显示英文标题

标题： Exact Sum Rules and Zeta Generating Formulas from the ODE/IM correspondence

摘要： 我们根据ODE/IM对应框架中$A_{2M-1}$T系统融合关系，为由${\cal PT}$对称势$V_{\cal PT}(x) = x^{2K} (ix)^\varepsilon$与$K,\varepsilon \in \mathbb{N}$定义的量子力学，以及由厄米势$V_{\cal H}(x) = x^{2 K}$与$K \in {\mathbb N} + 1$定义的量子力学，制定了精确求和规则（ESRs）和ζ生成公式（ZGFs）。 在此设置中，可积模型（IM）侧的融合关系对应于常微分方程（ODE）侧量化条件之间的递推关系，我们将其重新表述为谱泽塔函数（SZFs），$\zeta_n(s) = \sum_{\alpha \in {\mathbb N}_0} E_{n,\alpha}^{-s}$，其中$n$表示融合标签。 ESRs在固定$n$时产生SZFs之间的代数关系，而ZGFs则在不同融合标签的SZFs之间建立显式的函数映射。 这些结构由一个选择规则控制，该规则依赖于$M$和$n$，通常由融合关系中出现的切比雪夫多项式的结构和$\mathbb{Z}_{2M+2}$的西曼尼克旋转对称性的组合所引起。 我们的结果为可积模型中的T系统提供了新颖的谱解释，并指向了控制$\mathcal{PT}$对称性和厄米量子力学中全局谱数据的隐藏代数结构。 显示英文摘要

摘要： We formulate exact sum rules (ESRs) and zeta generating formulas (ZGFs) for quantum mechanics defined by the ${\cal PT}$-symmetric potential $V_{\cal PT}(x) = x^{2K} (ix)^\varepsilon$ with $K,\varepsilon \in \mathbb{N}$, as well as the Hermitian potential $V_{\cal H}(x) = x^{2 K}$ with $K \in {\mathbb N} + 1$, based on the fusion relations of the $A_{2M-1}$ T-system in the framework of the ODE/IM correspondence. In this setup, the fusion relations on the integrable model (IM) side correspond to recurrence relations among quantization conditions on the ordinary differential equation (ODE) side, which we reformulate in terms of spectral zeta functions (SZFs), $\zeta_n(s) = \sum_{\alpha \in {\mathbb N}_0} E_{n,\alpha}^{-s}$, where $n$ denotes the fusion label. The ESRs yield algebraic relations among SZFs at fixed $n$, while the ZGFs establish explicit functional mappings between SZFs at different fusion labels. These structures are governed by a selection rule depending on both $M$ and $n$, induced in general by the combination of the structure of the Chebyshev polynomials appearing in the fusion relations and the $\mathbb{Z}_{2M+2}$ Symanzik rotational symmetry. Our results provide a novel spectral interpretation of the T-system in integrable models and point toward hidden algebraic structures governing global spectral data in $\mathcal{PT}$-symmetric and Hermitian quantum mechanics.