This paper explores the complexity associated with solving the inverse Sturm-Liouville problem with Robin boundary conditions: given a sequence of eigenvalues and a sequence of norming constants, how many limits does a universal algorithm require to return the potential and boundary conditions? It is shown that if all but finitely many of the eigenvalues and norming constants coincide with those for the zero potential then the number of limits is zero, i.e. it is possible to retrieve the potential and boundary conditions precisely in finitely many steps. Otherwise, it is shown that this problem requires a single limit; moreover, if one has a priori control over how much the eigenvalues and norming constants differ from those of the zero-potential problem, and one knows that the average of the potential is zero, then the computation can be performed with complete error control. This is done in the spirit of the Solvability Complexity Index. All algorithms are provided explicitly along with numerical examples.

翻译：本文探讨了在Robin边界条件下求解逆Sturm-Liouville问题所涉及的复杂性：给定一个特征值序列和一个范数常数序列，通用算法需要经过多少次极限运算才能返回势函数和边界条件？研究表明，如果除有限个特征值和范数常数外，其余均与零势问题的特征值和范数常数一致，则所需极限次数为零，即可在有限步内精确恢复势函数和边界条件。否则，该问题需要一次极限运算；此外，若能够先验地控制特征值和范数常数与零势问题对应值的偏差程度，并且已知势函数的平均值为零，则可在完全误差控制下完成计算。这一研究遵循可解性复杂性指数的思想。本文明确给出了所有算法，并辅以数值示例进行说明。