ODE/IM correspondence for affine Lie algebras: a numerical approach

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.

Automated summary

In this study, the ODE/IM correspondence for untwisted affine Lie algebras associated with simple Lie algebras was investigated numerically, revealing that Q-functions in integrable models are expressed as the inner product of solutions of the dual linear problem. The Cheng's algorithm was used to efficiently determine the zeros of the Q-function, which are known to provide solutions of the Bethe ansatz equations. The results from numerical calculations were found to agree with those obtained from non-linear integral equations for simply-laced affine Lie algebras, including the exceptional type.