We present a comprehensive treatment of the non-periodic trigonometric $s\ell (2)$ Gaudin model with triangular boundary, with an emphasis on specific freedom found in the local realization of the generators, as well as in the creation operators used in the algebraic Bethe ansatz. First, we give Bethe vectors of the non-periodic trigonometric $s\ell (2)$ Gaudin model both through a recurrence relation and in a closed form. Next, the off-shell action of the generating function of the trigonometric Gaudin Hamiltonians with general boundary terms on an arbitrary Bethe vector is shown, together with the corresponding proof based on mathematical induction. The action of the Gaudin Hamiltonians is given explicitly. Furthermore, by careful choice of the arbitrary functions appearing in our more general formulation, we additionally obtain: i) the solutions to the Knizhnik-Zamolodchikov equations (each corresponding to one of the Bethe states); ii) compact formulas for the on-shell norms of Bethe states; and iii) closed-form expressions for the off-shell scalar products of Bethe states.

我们提出了非周期三角函数的综合处理 $秒\ell (2)$具有三角形边界的 Gaudin 模型，强调在生成器的局部实现中发现的特定自由，以及在代数 Bethe ansatz 中使用的创建运算符。首先，我们给出非周期三角函数的 Bethe 向量$秒\ell (2)$Gaudin 模型通过递推关系和封闭形式。接下来，展示了在任意 Bethe 向量上具有一般边界项的三角高丹哈密顿量的生成函数的壳外作用，以及基于数学归纳法的相应证明。Gaudin Hamiltonians 的作用是明确给出的。此外，通过仔细选择出现在我们更一般的公式中的任意函数，我们还获得了： i) Knizhnik-Zamolodchikov 方程的解（每个方程对应于 Bethe 状态之一）；ii) Bethe 状态的壳上范数的紧凑公式；和 iii) Bethe 状态的壳外标量积的闭式表达式。