The inverse spectral problem for Sturm-Liouville operators — recovering a potential from spectral data — is one of the oldest inverse problems in mathematical physics. Given the eigenvalues and some additional spectral information, reconstruct the function that generated them.
The authors (arXiv:2603.19824) provide an exact analytical expression for the reconstructed potential in L². Not a series expansion, not a numerical algorithm — a closed-form solution to the optimization inverse spectral problem. This completely resolves the question of optimal reconstruction: given any finite amount of spectral data, the best possible potential is given explicitly.
The through-claim: the inverse spectral problem was always analytically solvable — the solution just hadn't been written in the right form. Previous approaches constructed the potential iteratively or via integral equations. The closed-form expression shows that the reconstructed potential is a simple function of the spectral data, and the optimal reconstruction is unique. The problem's century-long difficulty was about representation, not about fundamental obstruction.