We derive the thermodynamic Bethe ansatz (TBA) equations for the Schrödinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann–Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr–Sommerfeld quantization condition.

中文翻译：

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具有规则奇点的Schrödinger方程的TBA方程

我们推导具有任意多项式电势和规则奇异（单极和双极）项的Schrödinger方程的热力学Bethe ansatz（TBA）方程。TBA方程提供了ODE / IM对应关系的非平凡泛化，并且提供了精确WKB方法中的Riemann-Hilbert问题的解决方案。我们将详细研究线性和谐波振荡器电势的TBA方程，以及逆和离心项。作为应用，我们还使用Bohr-Sommerfeld量化条件对这些电势的Voros频谱进行数值计算。