The nonlocal Darboux transformation for the time-independent axially symmetric Schrödinger and Helmholtz equations has been considered. New examples of two-dimensional potentials and exact solutions of the time-independent axially symmetric Schrödinger and Helmholtz equations have been obtained on the basis of the formulas of the nonlocal Darboux and generalized Moutard transformations.

中文翻译：

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关于时间无关轴对称薛定谔方程和亥姆霍兹方程的非局部达布变换

已经考虑了与时间无关的轴对称薛定谔方程和亥姆霍兹方程的非局部 Darboux 变换。基于非局部 Darboux 和广义 Moutard 变换的公式，已经获得了二维势的新例子和与时间无关的轴对称 Schrödinger 和 Helmholtz 方程的精确解。