In this paper, the authors investigate a partial inverse spectral problem for Sturm–Liouville operators with frozen arguments on star-shaped graphs. The problem is to reconstruct the potential on one edge from the known potentials on the other edges together with two sequences of eigenvalues from a prescribed spectral set. The proposed approach is constructive. First, the characteristic function associated with the given spectral data is constructed, allowing the unknown potential contribution to be isolated. The potential is then recovered by expanding the resulting expressions in an appropriate Riesz basis and solving a corresponding system of linear equations. Based on established uniqueness results, this procedure yields a constructive numerical algorithm. Numerical examples demonstrate reliable reconstruction for both smooth and piecewise continuous potentials, providing a practical scheme for frozen-argument problems on star graphs.

We consider the inverse resonance problem in scattering theory. In one-dimensional setting, the scattering matrix consists of $2\times 2$ entries of meromorphic functions. The resonances are defined as the poles of the meromorphic determinant. For the compactly supported perturbation, we are able to quantitatively estimate the zeros and poles of each meromorphic entry. The size of potential support is connected to the zero density of scattered wave field due to the form of Fourier transform. We will investigate certain properties of Fourier transforms in scattering theory and derive the inverse uniqueness on scattering source given certain knowledge on the perturbation and all the given resonances.

This research was devoted to investigate the inverse spectral problem of Sturm-Liouville operator with many frozen arguments. Under some assumptions, the authors obtained uniqueness theorems. At the end, a numerical simulation for the inverse problem was presented.