We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of N particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large N, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable β (β is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as N → ∞

中文翻译：

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Ising 模型的热力学平均值和 Toeplitz 矩阵的谱不变量

我们推导出一个通用公式，以某个无限托普利茨矩阵的有限子矩阵的谱不变量的显式定义函数的形式，给出 N 个粒子系统的一维 Ising 模型的配分函数的表示。我们获得了大 N 的配分函数的渐近表示，这可以作为明确计算一些热力学平均值的基础，例如，在一般平移不变自旋相互作用的情况下（不一定只在最近的自旋相互作用）邻居）。我们在复变量 β（β 是逆温度）的平面内从上方和下方估计分配函数，并考虑这些估计渐近等效为 N → ∞ 的条件