Partial Inverse Problems for the Sturm–Liouville Operator on a Star-Shaped Graph with Different Edge Lengths

The Sturm–Liouville operator with singular potentials from the class $W_2^{-1}$ is considered on a star-shaped graph with different edge lengths. We suppose that the potentials are known a priori on a part of the edges, and study the partial inverse problems, that consist in recovering the potentials on the other edges from a fractional part of the spectrum and some additional data. We prove three uniqueness theorems and provide a constructive algorithm for the solution of the inverse problems.