  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.
