In this paper, we for the first time get constructive solution for the inverse Sturm–Liouville problem with complex-valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The uniqueness of recovering the potential and the polynomials from the Weyl function is proved. An algorithm of solving the inverse problem is obtained and justified. More concretely, we reduce the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences and then derive reconstruction formulas for the problem coefficients, which are new even for the case of regular potential. Note that the spectral problem in this paper is investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. In the future, our results can be applied to investigation of the inverse problem solvability and stability as well as to development of numerical methods for the reconstruction.

在本文中，我们首次得到了具有复值奇异势和边界条件下谱参数多项式的Sturm-Liouville反问题的建设性解。证明了从Weyl函数恢复势能和多项式的唯一性。给出了求解反问题的算法并证明了其合理性。更具体地说，我们将非线性逆问题简化为有界无限序列的 Banach 空间中的线性方程，然后推导问题系数的重构公式，即使对于正则势的情况也是新的。请注意，本文中的谱问题是以一般非自伴形式研究的，并且我们的方法不要求谱的简单性。将来，