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Abstract

We study the inverse Sturm-Liouville problem on a finite interval from partial knowledge of spectral data. Specifically, we show that the potential can be uniquely reconstructed from the knowledge of a fraction of Dirichlet eigenvalues together with the normal derivatives of the corresponding eigenfunctions at both endpoints. We present two novel applications of our spectral result in inverse coefficient determination problems for evolutionary PDEs that include passive wave-based imaging of a medium and active imaging for the time-dependent Schr\"odinger equation with unknown internal sources. Our results yield optimal time measurement bounds for such inverse coefficient determination problems. A central innovation is the use of Kahane's interpolation theorem to analyze endpoint time traces of solutions, enabling the recovery without requiring analyticity assumptions or infinite-time data, as in previous approaches. Finally, in the appendix, we present a spectral interpolation theorem for one-dimensional Schr\"odinger operators, which may be of independent interest.