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For a selfadjoint Schr\"odinger operator on the half line with a real-valued,integrable, and compactly-supported potential, it is investigated whether theboundary parameter at the origin and the potential can uniquely be determinedby the scattering matrix or by the absolute value of the Jost function known atpositive energies, without having the bound-state information. It is provedthat, except in one special case where the scattering matrix has no boundstates and its value is $+1$ at zero energy, the determination by thescattering matrix is unique. In the special case, it is shown that there areexactly two distinct sets consisting of a potential and a boundary parameteryielding the same scattering matrix, and a characterization of thenonuniqueness is provided. A reconstruction from the scattering matrix isoutlined yielding all the corresponding potentials and boundary parameters. Theconcept of "eligible resonances" is introduced, and such resonances correspondto real-energy resonances that can be converted into bound states via a Darbouxtransformation without changing the compact support of the potential. It isproved that the determination of the boundary parameter and the potential bythe absolute value of the Jost function is unique up to the inclusion ofeligible resonances. Several equivalent characterizations are provided todetermine whether a resonance is eligible or ineligible. A reconstruction fromthe absolute value of the Jost function is given, yielding all thecorresponding potentials and boundary parameters. The results obtained areillustrated with various explicit examples.

Inverse problems for selfadjoint Schr\"odinger operators on the half line with compactly-supported potentials