We consider Sturm–Liouville problems with a discontinuity in an interior point, which are motivated by the inverse problems for the torsional modes of the Earth. We assume that the potential on the right half-interval and the coefficient in the right boundary condition are given. Half-inverse problems are studied, that consist in recovering the potential on the left half-interval and the left boundary condition from the eigenvalues. If the discontinuity belongs to the left half-interval, the position and the parameters of the discontinuity also can be reconstructed. In this paper, we provide reconstructing algorithms and prove existence of solutions for the considered inverse problems. Our approach is based on interpolation of entire functions.

中文翻译：

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不连续的Hochstadt-Lieberman问题的重构和可解性

我们考虑内点不连续的Sturm-Liouville问题，这是由地球扭转模态的反问题引起的。我们假设给出了右半间隔的电位和右边界条件下的系数。研究了半反问题，其中包括从特征值中恢复左半间隔和左边界条件的势。如果不连续点属于左半间隔，则也可以重新构造不连续点的位置和参数。在本文中，我们提供了重构算法并证明了所考虑的逆问题解的存在。我们的方法基于整个函数的插值。