We study numerically the ODE/IM correspondence for untwisted affine Lie algebras associated with simple Lie algebras including exceptional type. We consider the linear problem obtained from the massless limit of that of the modified affine Toda field equation. We found that the Q-functions in integrable models are expressed as the inner product of the solution of the dual linear problem and the subdominant solution of the linear problem. Using Cheng’s algorithm to obtain the solution of the linear problem, we can determine efficiently the zeros of the Q-function, which is known to provide the solutions of the Bethe ansatz equations (BAEs). We calculate the zeros numerically, which are shown to agree with the results from the non-linear integral equations (NLIEs) for simply-laced affine Lie algebras including the exceptional type. By the folding procedure of the Dynkin diagrams of simply-laced Lie algebras, we also find the correspondence for the linear problem of the non-simply-laced affine Lie algebras.

我们在数值上研究了与包括异常类型在内的简单李代数相关的非扭曲仿射李代数的 ODE/IM 对应关系。我们考虑从修正仿射 Toda 场方程的无质量极限得到的线性问题。我们发现可积模型中的Q函数表示为对偶线性问题的解和线性问题的次优解的内积。使用 Cheng 的算法获得线性问题的解，我们可以有效地确定Q的零点-函数，已知它可以提供 Bethe ansatz 方程 (BAE) 的解。我们以数值方式计算零点，这与包括异常类型在内的简单仿射李代数的非线性积分方程 (NLIE) 的结果一致。通过对单带李代数的Dynkin图的折叠过程，我们还找到了非单带仿射李代数的线性问题的对应关系。